{VERSION 7 1 "Linux" "7.1" } {USTYLETAB {PSTYLE "Ordered List 5" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 } {PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "MS Serif" 1 12 0 0 0 1 1 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output " -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 2 2 1 3 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "MS S erif" 1 14 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 12 40 120 40 1 2 2 2 2 2 1 2 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered \+ List 4" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 2 2 1 2 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "H eading 2" -1 4 1 {CSTYLE "" -1 -1 "MS Serif" 1 16 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courie r" 1 12 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "MS Serif" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered Li st 2" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "Equation Label" -1 200 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 201 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 1 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "MS Serif" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 0 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 202 "Couri er" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "MS Ser if" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{PSTYLE "" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 203 38 "Bevezet\303\251s a matem atik\303\241ba" }}}{EXCHG {PARA 19 "" 0 "" {TEXT 204 18 "J\303\241rai \+ Antal" }}}{EXCHG {PARA 19 "" 0 "" {TEXT 204 68 "Ezek a programok csak \+ szeml\303\251ltet\303\251sre szolg\303\241lnak" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 205 11 "1. Halmazok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 19 "1.1. Logikai alapok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 23 "1.1.1. Az axiomatikus m" }{TEXT 207 8 "\303\263" }{TEXT 207 6 "dszer." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 59 "1.1.2. Logikai jelek, predik\303\241tumok, formul\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 14 "Logikai jelek :" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "\"¬∧∨⇒⇔⊕|‖\"; 1;" "C$ QK¬∧∨⇒⇔⊕|gr‖6\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "QK¬∧∨⇒⇔⊕|gr‖6\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{PARA 0 "" 0 "" {TEXT 201 10 "Kvantorok:" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "\"∃∀\"; 1;" "C$Q0∃ ∀6\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "Q0∃∀6 \"" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 41 " Formul\303\241k gener\303\241l\303\241sa:" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "ppnot := \"¬\"; -1; ppand := \"∧\"; -1; ppor := \"∨\"; -1; ppimply := \"&Impli es;\"; -1; ppiff := \"⇔\"; -1; ppexist := \"∃\"; -1; ppforal l := \"∀\"; -1;" "C0>I&ppnotG6\"Q&¬F%!\"\">I&ppandGF%Q&&and ;F%F'>I%pporGF%Q%∨F%F'>I(ppimplyGF%Q*⇒F%F'>I&ppiffGF%Q&&iff ;F%F'>I(ppexistGF%Q(∃F%F'>I)ppforallGF%Q)∀F%F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "stringseq:=proc() local i,s; \n" }{MPLTEXT 1 0 61 "if nargs=0 then return \"\" fi; s:=convert(args[ 1],string);\n" }{MPLTEXT 1 0 62 "for i from 2 to nargs do s:=cat(s,\", \",args[i]) od; s; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6\"6$I\"i GF#I\"sGF#F#F#C&@$/%&nargsG\"\"!OQ!F#>F&-I(convertG%*protectedG6$&%%ar gsG6#\"\"\"I'stringGF1?(F%\"\"#F6F*I%trueGF1>F&-I$catGF16%F&Q\",F#&F46 #F%F&F#F#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(combina t):\n" }{MPLTEXT 1 0 51 "genform:=proc(L::list(list(string)),n::nonneg int)\n" }{MPLTEXT 1 0 37 "local V,VV,i,j,c,a,C,S,SL,SR,sr,sl;\n" } {MPLTEXT 1 0 54 "if nops(L)<=1 then return \{\} fi; S:=\{\}; V:=L[1]; \n" }{MPLTEXT 1 0 48 "if n=0 then VV:=[]; for i from 2 to nops(L) do\n " }{MPLTEXT 1 0 53 " C:=choose(VV,i-2); for c in C do for a in L[i] d o\n" }{MPLTEXT 1 0 57 " S:=S union \{cat(a,\"(\",stringseq(op(c)), \")\")\};\n" }{MPLTEXT 1 0 35 " od; od; VV:=[op(VV),op(V)]; od;\n" } {MPLTEXT 1 0 42 "else SL:=genform(L,n-1); C:=choose(V,1);\n" }{MPLTEXT 1 0 61 "for sl in SL do S:=S union \{cat(ppnot,sl)\}; for c in C do\n " }{MPLTEXT 1 0 57 " S:=S union \{cat(\"(\",ppexist,op(c),\" \",sl,\" )\"),\n" }{MPLTEXT 1 0 46 " cat(\"(\",ppforall,op(c),\" \",sl,\")\") \}\n" }{MPLTEXT 1 0 9 "od; od;\n" }{MPLTEXT 1 0 55 "for j to n do SL:= genform(L,j-1); SR:=genform(L,n-j);\n" }{MPLTEXT 1 0 35 " for sl in S L do for sr in SR do\n" }{MPLTEXT 1 0 76 " S:=S union \{cat(\"(\",s l,ppand,sr,\")\"),cat(\"(\",sl,ppor,sr,\")\"),\n" }{MPLTEXT 1 0 67 " \+ cat(\"(\",sl,ppimply,sr,\")\"),cat(\"(\",sl,ppiff,sr,\")\")\}\n" } {MPLTEXT 1 0 11 " od; od;\n" }{MPLTEXT 1 0 17 "od; fi; S; end;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"LG6\"-I%listG%*protectedG6#-F(6# I'stringGF)'I\"nGF&I*nonnegintGF)6.I\"VGF&I#VVGF&I\"iGF&I\"jGF&I\"cGF& I\"aGF&I\"CGF&I\"SGF&I#SLGF&I#SRGF&I#srGF&I#slGF&F&F&C'@$1-I%nopsGF)6# F%\"\"\"O<\">F9FF>F2&F%6#FD@%/F/\"\"!C$>F37\"?(F4\"\"#FDFAI%trueGF)C%> F8-_I)combinatG6$F)I(_syslibGF&I'chooseGF&6$F3,&F4FDFR!\"\"?&F6F8FS?&F 7&F%6#F4FS>F9-I&unionGF)6$F9<#-I$catGF)6&F7Q\"(F&-I*stringseqGF&6#-I#o pGF)6#F6Q\")F&>F37$-Fjo6#F3-Fjo6#F2C&>F:-I(genformGF&6$F%,&F/FDFDFhn>F 8-FW6$F2FD?&F=F:FSC$>F9-F_o6$F9<#-Fco6$I&ppnotGF&F=?&F6F8FS>F9-F_o6$F9 <$-Fco6(FeoI(ppexistGF&FioQ\"~F&F=F\\p-Fco6(FeoI)ppforallGF&FioF]rF=F \\p?(F5FDFDF/FSC%>F:-Ffp6$F%,&F5FDFDFhn>F;-Ffp6$F%,&F/FDF5Fhn?&F=F:FS? &FF9-F_o6$F9<&-Fco6'FeoF=I&ppandGF&F " 0 "" {MPLTEXT 1 0 71 "genform([[\"x\",\"y\"],[\" A\",\"B\"],[\"C\",\"D\"],[\"E\",\"F\"]],0);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<.Q$A()6\"Q$B()F$Q%C(x)F$Q%D(x)F$Q%C(y)F$Q%D(y)F$Q'E(x,x) F$Q'F(x,x)F$Q'E(x,y)F$Q'F(x,y)F$Q'E(y,y)F$Q'F(y,y)F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "genform([[\"x\",\"y\"],[],[\"P\",\"E\"],[ \"I\"]],1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " " 0 "" {MPLTEXT 1 0 46 "genform([[\"x\",\"y\"],[],[\"P\"],[\"I\"]],2):" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 467 "Az al\303\241bbi parser program egy sztring elemz\303\2 51s\303\251t v\303\251gzi, hogy az \303\251rv\303\251nyes formula-e? H a igen, akkor a true \303\251rt\303\251ket \303\251s p\303\241rok egy \+ list\303\241j\303\241t, \303\251s a marad\303\251k sztringet adja viss za. A p\303\241rok m\303\241sodik koordin\303\241t\303\241ja a megtal \303\241lt szintaktikai alapegys\303\251g. Az els\305\221 koordin\303 \241ta az alapegys\303\251g tipusa, az al\303\241bbiak szerint:" }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 16 "k \+ kvantor" }}{PARA 0 "" 0 "" {TEXT 201 21 "l logikai jel" }} {PARA 0 "" 0 "" {TEXT 201 30 "p z\303\241r\303\263jel" }}{PARA 0 "" 0 "" {TEXT 201 23 "c vessz\305\221" }}{PARA 0 "" 0 "" {TEXT 201 38 "f szabad v\303\241ltoz\303\263" }}{PARA 0 "" 0 " " {TEXT 201 52 "b k\303\266t\303\266tt v\303\241ltoz\303\263" }}{PARA 0 "" 0 "" {TEXT 201 79 "0,1,... predik\303\241tum, adott sz \303\241m\303\272 v\303\241ltoz\303\263val" }}{PARA 0 "" 0 "" {TEXT 201 95 "? predik\303\241tum, m\303\251g ismeretlen sz\303\241m \303\272 v\303\241ltoz\303\263val" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 131 "Hiba e set\303\251n false \303\251rt\303\251ket kapunk, a lista \303\251s a m arad\303\251k sztring pedig utal a hiba hely\303\251re." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 323 "A v\303\241ltoz\3 03\263k \303\251s a predik\303\241tumok neve bet\305\261vel kell kezd \305\221dj\303\266n \303\251s bet\305\261ket \303\251s sz\303\241mjegy eket tartalmazhat. Elv\303\241laszt\303\263 jelk\303\251nt tetsz\305\2 21leges \"whitespace\" karakterekb\305\221l \303\241ll\303\263 sorozat haszn\303\241lhat\303\263." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 63 "Az egyszer\30 5\261bb elj\303\241r\303\241sokkal kezdj\303\274k:" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 285 "A parseparentheses elj \303\241r\303\241s egy kezd\305\221 z\303\241r\303\263jellel kezd\305 \221d\305\221 sztringben megkeresi az ehhez tartoz\303\263 z\303\241r \303\263 z\303\241r\303\263jelet, \303\251s ennek index\303\251t adja \+ vissza. Ha sikertelen, akkor nulla az eredm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 118 "A parsename elj\3 03\241r\303\241s egy sztringet sz\303\251tv\303\241g egy n\303\251vre \+ \303\251s egy marad\303\251kra." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {PARA 0 "" 0 "" {TEXT 201 124 "A parsevarsec elj\303\241r\303\241s ves sz\305\221kel elv\303\241lasztott v\303\241ltoz\303\263k sorozat\303\2 41t ismeri fel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 73 "A parsepredicate elj\303\241r\303\241s egy predik\303\24 1tumot ismer fel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 57 "A f\305\221program a pa rsesentence elj\303\241r\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" } }{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "parseparentheses:=proc(s::string) local n,j;\n" }{MPLTEXT 1 0 35 "if s=\"\" then return 0 fi; n:=0;\n" }{MPLTEXT 1 0 23 "for j to le ngth(s) do\n" }{MPLTEXT 1 0 33 " if s[j]=\"(\" then n:=n+1 fi;\n" } {MPLTEXT 1 0 34 " if s[j]=\")\" then n:=n-1; fi;\n" }{MPLTEXT 1 0 28 " if n=0 then return j fi;\n" }{MPLTEXT 1 0 13 "od; 0; end;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"sG6\"I'stringG%*protectedG6$I\"n GF&I\"jGF&F&F&C&@$/F%Q!F&O\"\"!>F*F1?(F+\"\"\"F4-I'lengthGF(6#F%I%true GF(C%@$/&F%6#F+Q\"(F&>F*,&F*F4F4F4@$/FF*,&F*F4F4!\"\"@$/F*F1OF +F1F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(StringToo ls):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "parsename:=proc(s ::string) local ls,rs;\n" }{MPLTEXT 1 0 18 "rs:=TrimLeft(s);\n" } {MPLTEXT 1 0 43 "if length(rs)=0 then return \"\",\"\" fi;\n" } {MPLTEXT 1 0 74 "if IsAlpha(rs[1]) then ls:=rs[1]; rs:=Drop(rs,1) else return \"\",rs fi;\n" }{MPLTEXT 1 0 23 "while length(rs)>0 do\n" } {MPLTEXT 1 0 33 " if IsAlphaNumeric(rs[1]) then\n" }{MPLTEXT 1 0 40 " ls:=cat(ls,rs[1]); rs:=Drop(rs,1);\n" }{MPLTEXT 1 0 25 " else ret urn ls,rs fi;\n" }{MPLTEXT 1 0 16 "od; ls,rs end;\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "f*6#'I\"sG6\"I'stringG%*protectedG6$I#lsGF&I#rsGF&F&F&C '>F+-_I,StringToolsG6$F(I(_syslibGF&I)TrimLeftGF&6#F%@$/-I'lengthGF(6# F+\"\"!O6$Q!F&F=@%-_F0I(IsAlphaGF&6#&F+6#\"\"\"C$>F*FC>F+-_F0I%DropGF& 6$F+FEO6$F=F+?(F&FEFEF&2F:F7@%-_F0I/IsAlphaNumericGF&FBC$>F*-I$catGF(6 $F*FCFHOF)F)F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "parse varseq:=proc(s::string) local L,rs,x;\n" }{MPLTEXT 1 0 64 "rs:=TrimLef t(s); L:=[]; if rs=\"\" then return true,L,\"\" fi;\n" }{MPLTEXT 1 0 15 "while true do\n" }{MPLTEXT 1 0 21 " x:=parsename(rs);\n" } {MPLTEXT 1 0 45 " if x[1]=\"\" then return false,L,x[2] fi;\n" } {MPLTEXT 1 0 29 " L:=[op(L),[\"f\",x[1]]]; \n" }{MPLTEXT 1 0 62 " rs :=TrimLeft(x[2]); if rs=\"\" then return true,L,\"\" fi;\n" }{MPLTEXT 1 0 46 " if rs[1]<>\",\" then return false,L,rs fi;\n" }{MPLTEXT 1 0 29 " L:=[op(L),[\"c\",\",\"]];\n" }{MPLTEXT 1 0 29 " rs:=TrimLeft(Dr op(rs,1));\n" }{MPLTEXT 1 0 43 " if rs=\"\" then return false,L,\"\" \+ fi;\n" }{MPLTEXT 1 0 10 "od; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f *6#'I\"sG6\"I'stringG%*protectedG6%I\"LGF&I#rsGF&I\"xGF&F&F&C&>F+-_I,S tringToolsG6$F(I(_syslibGF&I)TrimLeftGF&6#F%>F*7\"@$/F+Q!F&O6%I%trueGF (F*F:?(F&\"\"\"F?F&F=C+>F,-I*parsenameGF&6#F+@$/&F,6#F?F:O6%I&falseGF( F*&F,6#\"\"#>F*7$-I#opGF(6#F*7$Q\"fF&FG>F+-F06#FLF8@$0&F+FHQ\",F&O6%FK F*F+>F*7$FQ7$Q\"cF&Ffn>F+-F06#-_F1I%DropGF&6$F+F?@$F9O6%FKF*F:F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "parsepredicate:=proc(s::s tring) local L,x,ls,rs,j;\n" }{MPLTEXT 1 0 27 "x:=parsename(s);ls:=x[1 ];\n" }{MPLTEXT 1 0 52 "if ls=\"\" then return false,[],x[2] fi; rs:=x [2];\n" }{MPLTEXT 1 0 50 "if rs=\"\" then return false,[\"?\",ls],\"\" fi;\n" }{MPLTEXT 1 0 53 "if rs[1]<>\"(\" then return false,[\"?\",ls] ,rs fi;\n" }{MPLTEXT 1 0 26 "j:=parseparentheses(rs);\n" }{MPLTEXT 1 0 44 "if j=0 then return false,[\"?\",ls],rs fi;\n" }{MPLTEXT 1 0 29 " x:=parsevarseq(rs[2..j-1]);\n" }{MPLTEXT 1 0 14 "if x[1] then\n" } {MPLTEXT 1 0 19 " if x[2]=[] then\n" }{MPLTEXT 1 0 58 " true,[[0,l s],[\"p\",\"(\"],[\"p\",\")\"]],Drop(rs,j)\n" }{MPLTEXT 1 0 8 " else \n" }{MPLTEXT 1 0 72 " true,[[(nops(x[2])+1)/2,ls],[\"p\",\"(\"],op (x[2]),[\"p\",\")\"]],\n" }{MPLTEXT 1 0 18 " Drop(rs,j)\n" } {MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 46 "else x[1],x[2],cat(x[3],Dro p(rs,j-1)) fi; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"sG6\"I'st ringG%*protectedG6'I\"LGF&I\"xGF&I#lsGF&I#rsGF&I\"jGF&F&F&C,>F+-I*pars enameGF&6#F%>F,&F+6#\"\"\"@$/F,Q!F&O6%I&falseGF(7\"&F+6#\"\"#>F-F?@$/F -F:O6%F=7$Q\"?F&F,F:@$0&F-F6Q\"(F&O6%F=FGF->F.-I1parseparenthesesGF&6# F-@$/F.\"\"!FM>F+-I,parsevarseqGF&6#&F-6#;FA,&F.F7F7!\"\"@%F5@%/F?F>6% I%trueGF(7%7$FUF,7$Q\"pF&FL7$FaoQ\")F&-_I,StringToolsG6$F(I(_syslibGF& I%DropGF&6$F-F.6%F]o7&7$,&*&#F7FAF7-I%nopsGF(6#F?F7F7F`pF7F,F`o-I#opGF (FcpFboFdo6%F5F?-I$catGF(6$&F+6#\"\"$-Feo6$F-FgnF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "parsesentence:=proc(s::string) loca l ls,rs,L,x,j,n;\n" }{MPLTEXT 1 0 59 " global ppnot,ppand,ppor,ppimpl y,ppiff,ppexist,ppforall;\n" }{MPLTEXT 1 0 59 "rs:=TrimLeft(s); if rs= \"\" then return false,[],\"\" fi;\n" }{MPLTEXT 1 0 28 "if IsPrefix(pp not,rs) then\n" }{MPLTEXT 1 0 31 " rs:=Drop(rs,length(ppnot));\n" } {MPLTEXT 1 0 25 " x:=parsesentence(rs);\n" }{MPLTEXT 1 0 38 " x[1],[ [\"l\",ppnot],op(x[2])],x[3]\n" }{MPLTEXT 1 0 24 "elif rs[1]<>\"(\" th en\n" }{MPLTEXT 1 0 22 " parsepredicate(rs)\n" }{MPLTEXT 1 0 6 "else \n" }{MPLTEXT 1 0 48 " L:=[[\"p\",\"(\"]]; j:=parseparentheses(rs);\n " }{MPLTEXT 1 0 44 " if j=0 then return false,L,rs[2..-1] fi;\n" } {MPLTEXT 1 0 47 " ls:=TrimLeft(rs[2..j-1]); rs:=Drop(rs,j-1);\n" } {MPLTEXT 1 0 57 " if IsPrefix(ppexist,ls) or IsPrefix(ppforall,ls) th en\n" }{MPLTEXT 1 0 34 " if IsPrefix(ppexist,ls) then\n" }{MPLTEXT 1 0 65 " L:=[op(L),[\"k\",ppexist]]; ls:=Drop(ls,length(ppexist)) ;\n" }{MPLTEXT 1 0 10 " else\n" }{MPLTEXT 1 0 67 " L:=[op(L),[ \"k\",ppforall]]; ls:=Drop(ls,length(ppforall));\n" }{MPLTEXT 1 0 9 " \+ fi;\n" }{MPLTEXT 1 0 73 " x:=parsename(ls); if x[1]=\"\" then re turn false,L,cat(x[2],rs) fi;\n" }{MPLTEXT 1 0 35 " n:=x[1];L:=[op( L),[\"b\",n]];\n" }{MPLTEXT 1 0 29 " x:=parsesentence(x[2]);\n" } {MPLTEXT 1 0 36 " if not x[1] or x[3]<>\"\" then\n" }{MPLTEXT 1 0 54 " return false,[op(L),op(x[2])],cat(x[3],rs) fi;\n" }{MPLTEXT 1 0 22 " for j in x[2] do\n" }{MPLTEXT 1 0 37 " if j[1]=\"f\" \+ and j[2]=n then\n" }{MPLTEXT 1 0 35 " L:=[op(L),[\"b\",n]] else \n" }{MPLTEXT 1 0 30 " L:=[op(L),j] fi; od;\n" }{MPLTEXT 1 0 57 " L:=[op(L),[\"p\",\")\"]]; rs:=TrimLeft(Drop(rs,1));\n" } {MPLTEXT 1 0 15 " true,L,rs\n" }{MPLTEXT 1 0 8 " else\n" }{MPLTEXT 1 0 27 " x:=parsesentence(ls);\n" }{MPLTEXT 1 0 57 " if not x[1 ] then return false,x[2],cat(x[3],rs) fi;\n" }{MPLTEXT 1 0 46 " L:= [op(L),op(x[2])]; ls:=TrimLeft(x[3]);\n" }{MPLTEXT 1 0 48 " if IsPr efix(ppand,ls) or IsPrefix(ppor,ls)\n" }{MPLTEXT 1 0 60 " or Is Prefix(ppimply,ls) or IsPrefix(ppiff,ls) then\n" }{MPLTEXT 1 0 34 " \+ if IsPrefix(ppand,ls) then\n" }{MPLTEXT 1 0 61 " L:=[op(L),[ \"l\",ppand]];ls:=Drop(ls,length(ppand))\n" }{MPLTEXT 1 0 35 " el if IsPrefix(ppor,ls) then\n" }{MPLTEXT 1 0 59 " L:=[op(L),[\"l \",ppor]];ls:=Drop(ls,length(ppor))\n" }{MPLTEXT 1 0 35 " elif Is Prefix(ppor,ls) then\n" }{MPLTEXT 1 0 59 " L:=[op(L),[\"l\",ppo r]];ls:=Drop(ls,length(ppor))\n" }{MPLTEXT 1 0 12 " else\n" } {MPLTEXT 1 0 59 " L:=[op(L),[\"l\",ppor]];ls:=Drop(ls,length(pp or))\n" }{MPLTEXT 1 0 11 " fi;\n" }{MPLTEXT 1 0 49 " x:=pars esentence(ls);L:=[op(L),op(x[2])];\n" }{MPLTEXT 1 0 38 " if not x [1] or x[3]<>\"\" then\n" }{MPLTEXT 1 0 41 " return false,L,cat (x[3],rs) fi;\n" }{MPLTEXT 1 0 59 " L:=[op(L),[\"p\",\")\"]]; rs: =TrimLeft(Drop(rs,1));\n" }{MPLTEXT 1 0 17 " true,L,rs\n" } {MPLTEXT 1 0 10 " else\n" }{MPLTEXT 1 0 26 " false,L,cat(ls,rs )\n" }{MPLTEXT 1 0 9 " fi;\n" }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 8 "fi; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"sG6\"I'string G%*protectedG6(I#lsGF&I#rsGF&I\"LGF&I\"xGF&I\"jGF&I\"nGF&F&F&C%>F+-_I, StringToolsG6$F(I(_syslibGF&I)TrimLeftGF&6#F%@$/F+Q!F&O6%I&falseGF(7\" F;@'-_F4I)IsPrefixGF&6$I&ppnotGF&F+C%>F+-_F4I%DropGF&6$F+-I'lengthGF(6 #FE>F--I.parsesentenceGF&6#F+6%&F-6#\"\"\"7$7$Q\"lF&FE-I#opGF(6#&F-6# \"\"#&F-6#\"\"$0&F+FUQ\"(F&-I/parsepredicateGF&FRC(>F,7#7$Q\"pF&F_o>F. -I1parseparenthesesGF&FR@$/F.\"\"!O6%F>F,&F+6#;Fin!\"\">F*-F36#&F+6#;F in,&F.FVFVFbp>F+-FI6$F+Fip@%5-FB6$I(ppexistGF&F*-FB6$I)ppforallGF&F*C- @%F_qC$>F,7$-Fen6#F,7$Q\"kF&Faq>F*-FI6$F*-FM6#FaqC$>F,7$Fjq7$F]rFdq>F* -FI6$F*-FM6#Fdq>F--I*parsenameGF&6#F*@$/FTF;O6%F>F,-I$catGF(6$FgnF+>F/ FT>F,7$Fjq7$Q\"bF&F/>F--FQFfn@$54FT0FjnF;O6%F>7$FjqFZ-Fes6$FjnF+?&F.Fg nI%trueGF(@%3/&F.FUQ\"fF&/&F.FhnF/Fhs>F,7$FjqF.>F,7$Fjq7$FfoQ\")F&>F+- F36#-FI6$F+FV6%FhtF,F+C'>F--FQF_s@$F`tO6%F>FgnFet>F,Fdt>F*-F36#Fjn@%55 5-FB6$I&ppandGF&F*-FB6$I%pporGF&F*-FB6$I(ppimplyGF&F*-FB6$I&ppiffGF&F* C)@)FjvC$>F,7$Fjq7$FYF\\w>F*-FI6$F*-FM6#F\\wF]wC$>F,7$Fjq7$FYF_w>F*-FI 6$F*-FM6#F_wF]wFaxFaxF]vFbv@$F_tO6%F>F,FetFbuFfuF[v6%F>F,-Fes6$F*F+F&6 )FEF\\wF_wFbwFewFaqFdqF&" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "p arsesentence(\"I(x,y)\"); 1; parsesentence(\"∀x I(x,x)\"); 1; p arsesentence(\"(∀x I(x,x))\"); 1; parsesentence(\"(∀x(&f orall;y(∃z(I(x,z)∧I(y,z)))))\"); 1; parsesentence(\"((E(x)&a nd;E(y))∧¬I(x,y))\"); 1;" "C,-I.parsesentenceG6\"6#Q'I(x,y)F% \"\"\"-F$6#Q1∀x~I(x,x)F%F(-F$6#Q3(∀x~I(x,x))F%F(-F$6#QT( ∀x(∀y(∃z(I(x,z)∧I(y,z)))))F%F(-F$6#QB((E(x)&an d;E(y))∧¬I(x,y))F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%I%true G%*protectedG7(7$\"\"#Q\"I6\"7$Q\"pF)Q\"(F)7$Q\"fF)Q\"xF)7$Q\"cF)Q\",F )7$F.Q\"yF)7$F+Q\")F)Q!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%I&falseG% *protectedG7\"Q1∀x~I(x,x)6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %I%trueG%*protectedG7,7$Q\"p6\"Q\"(F(7$Q\"kF(Q)∀F(7$Q\"bF(Q\"xF (7$\"\"#Q\"IF(F&F-7$Q\"cF(Q\",F(F-7$F'Q\")F(F6Q!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%I%trueG%*protectedG7=7$Q\"p6\"Q\"(F(7$Q\"kF(Q)∀ F(7$Q\"bF(Q\"xF(F&F*7$F.Q\"yF(F&7$F+Q(∃F(7$F.Q\"zF(F&7$\"\"#Q\"I F(F&F-7$Q\"cF(Q\",F(F47$F'Q\")F(7$Q\"lF(Q&∧F(F6F&F0F9F4F1.1.13. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "- >1.1.14. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.1.15. \+ Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.1.16. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.1.17. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.1.18. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.1.19. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.1.20. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.1.21. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.1 .22. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "1.1.23. Tov" } {TEXT 207 8 "\303\241" }{TEXT 207 14 "bbi feladatok." }}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 39 "1.2. Halmazelm\303\251leti alapfogalmak" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 28 "1.2.1. Halmazelm\303\251let." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A:=\{a,b,c \}; member(b,A); member(d,A); b in A; evalb(%); d in A; evalb(%);" }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG 6\"I\"cGF$I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG " }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I#inG%*protectedG6$I\"bG6\"<%F&I\"cGF'I\"aGF'" }} {PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I#inG%*protectedG6$I\"dG6\"<%I\"bGF'I\"cGF'I\"aGF'" }} {PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "whattype(A); whattype(a); whattype(2); what type(krikszkraksz); whattype(\"krikszkraksz\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$setG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I'sym bolG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I(integerG%*protecte dG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I'symbolG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I'stringG%*protectedG" }}}}{SECT 0 {PARA 5 "" 0 " " {TEXT 207 38 "1.2.2. Meghat\303\241rozotts\303\241g." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "\{a,a,b,b,a\}; \{b,a\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"aG F$" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 55 "* 1.2.3. Meghat\303\241r ozotts\303\241gi axi\303\263ma." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 27 "1.2.4. R\303\251 szhalmazok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "\{a,b\} subset \{a,b,d\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "\{a,c\} subset \{a,b,d\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "I& falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "\{a,b \} subset \{a,b\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protected G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "X subset X; X subset Y ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'subsetG%*protectedG6$I\"XG6\"I\"YGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "A:=\{1,2,3,4,5,6,7\}; select(x->isp rime(x),A);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<)\"\"\"\"\"#\"\"$\"\"% \"\"&\"\"'\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&\"\"#\"\"$\"\"&\"\" (" }}}{PARA 0 "" 0 "" {TEXT 2 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 10 "* 1.2.5. R" }{TEXT 207 31 "\303\251szhalmaz-axi\303\263ma." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "* 1.2.6. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 27 "* 1. 2.7. Megjegyz\303\251s." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#% ?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 51 "1.2.8. N\303\251h\303\24 1ny egyszer\305\261 halmaz." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "\{\}; \{a,b\}; \{a,a\}; \{a, b,c\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#I\"a G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG6\"I\"cGF$I\"aGF$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a:=b; \{a,b\};" }}{PARA 11 " " 1 "" {XPPMATH 20 "I\"bG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#I\"bG6 \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a:='a'; \{a,b\};" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"aG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"aGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "\{\} subset X;" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}} {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 52 "1.2.9. Az \303\274res halmaz axi\303\263m\303\241ja." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 32 "1.2.10. P\303\241raxi\303\263ma." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 20 "1 .2.11. Uni\303\263." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`union`(\{b, a\}, \{b, c\}); 1; `union`(\{b, a\}, \{b, c\}); 1;" "C&-I&unionG%*protectedG6$<$I\"bG6\"I\"aGF)<$F(I \"cGF)\"\"\"F#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG6\"I\"cGF$I\" aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG6\"I\"cGF$I\"aGF$" }}} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`𝒜` := \{\{\}, \{b, a\}, \+ \{a\}, \{1, 2, b\}, \{b, c\}\}; 1; op(`𝒜`); 1; `union`(op(`𝒜 `)); 1; `union`(\{a\}, \{b, c\}, \{1, 2, b\}, \{\}, \{b, a\}); 1;" "C* >I'𝒜G6\"<'<\"<$I\"bGF%I\"aGF%<#F*<%\"\"\"\"\"#F)<$F)I\"cGF%F--I#o pG%*protectedG6#F$F--I&unionGF36#F1F--F66'F+F/F,F'F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'<\"<$I\"bG6\"I\"aGF&<#F'<%\"\"\"\"\"#F%<$F%I\"cGF& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6'<\"<$I\"bG6\"I\"aGF&<#F'<%\"\"\"\" \"#F%<$F%I\"cGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'\"\"\"\"\"#I\"bG6 \"I\"cGF&I\"aGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'\"\"\"\"\"#I\"bG6 \"I\"cGF&I\"aGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "\{a,b\} union \{b,c\}; `union`(\{a\},\{b,c\},\{1,2,b\},\{\},\{a,b\});\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG6\"I\"cGF$I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'\"\"\"\"\"#I\"bG6\"I\"cGF&I\"aGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=\{a,b\}; B:=\{b,c\}; `union`(A,B );" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"cGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<% I\"bG6\"I\"cGF$I\"aGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "` union`();" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "* 1.2.12. Uni \303\263axi\303\263ma." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%? G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 74 "1.2.13. \303\201ll\303\25 5t\303\241s: az uni\303\263 tulajdons\303\241gai." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "X union \{ \};" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"XG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "X union Y; Y union X;" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&unionG%*protectedG6$I\"XG6\"I\"YGF'" }}{PARA 11 "" 1 " " {XPPMATH 20 "-I&unionG%*protectedG6$I\"XG6\"I\"YGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "(X union Y) union Z; X union (Y uni on Z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&unionG%*protectedG6%I\"XG6 \"I\"YGF'I\"ZGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&unionG%*protected G6%I\"XG6\"I\"YGF'I\"ZGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "X union X;" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"XG6\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "1.2.14. Metszet." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "\{a,b,1\} intersect \{a,c,2, 1\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"I\"aG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`intersect`(\{a,b,c,d\},\{a,b,c,1\} ,\{a,b,1,2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"aGF$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "A:=\{a,b\}; B:=\{b,c\}; C:= \{c,a\}; A intersect B; B intersect C; C intersect A; `intersect`(A,B) ; `intersect`(A,B,C);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\" aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"cGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"cG6\"I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 " <#I\"bG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#I\"cG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#I\"aG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#I\"bG6 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 8 "1.2.1 5. " }{TEXT 207 61 "\303\201ll\303\255t\303\241s: a metszet tulajdons \303\241gai." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 66 "Az els\305\221 n\303\251gy tulajdons\303\241got a Maple is is meri:" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "X intersect \{\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "%# %?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "X intersect Y; Y int ersect X;" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*intersectG%*protectedG6$ I\"XG6\"I\"YGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*intersectG%*protec tedG6$I\"XG6\"I\"YGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "X \+ intersect (Y intersect Z); (X intersect Y) intersect Z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*intersectG%*protectedG6%I\"XG6\"I\"YGF'I\"ZGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*intersectG%*protectedG6%I\"XG6\"I\" YGF'I\"ZGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "X intersect \+ X;" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"XG6\"" }}}{EXCHG {PARA 0 "" 0 " " {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "-> 1.2.16. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.17. F eladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.18. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.19. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.1.20. \+ Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.2.21. Feladat." } }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 8 "1.2.22. " }{TEXT 207 72 "\303\2 01ll\303\255t\303\241s: disztributivit\303\241si szab\303\241lyok." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "X intersect (Y union Z); expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*intersectG%*protectedG6$-I&unionGF$6$I\"YG6\"I\"ZGF*I \"XGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&unionG%*protectedG6$-I*inte rsectGF$6$I\"XG6\"I\"YGF*-F'6$F)I\"ZGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "X union (Y intersect Z); A:=\{a,b,c\}; B:=\{b,c,d\}; C:=\{c,d,e\}; A union (B intersect C); (A union B) intersect (A union C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&unionG%*protectedG6$I\"XG6\"- I*intersectGF$6$I\"YGF'I\"ZGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"b G6\"I\"cGF$I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG6\"I\"cGF$I \"dGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"cG6\"I\"dGF$I\"eGF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"cGF$I\"aGF$I\"dGF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"cGF$I\"aGF$I\"dGF$" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.23. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 62 "1.2. 24. K\303\274l\303\266nbs\303\251g \303\251s komplementer." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A:= \{a,b\}; B:=\{b,c\}; C:=\{a,b,c,d\}; A minus B;\n" }{MPLTEXT 1 0 33 "s ymmdiff(A,B); symmdiff(A,B,C);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I \"bG6\"I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"cGF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"cGF$I\"aGF$I\"dGF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "<#I\"aG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"cG6\"I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"dG F$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 37 "1.2.25. \303\201ll\303\255t\303\241s." } }{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A; C minus (C minus A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG 6\"I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"aGF$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "C; C minus \{\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"cGF$I\"aGF$I\"dGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"cGF$I\"aGF$I\"dGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "\{\}, C minus C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<\"F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "\{ \}; A intersect (C minus A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" } }{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "C; A union (C minus A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"cGF$I\"aGF$I\"dGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "< &I\"bG6\"I\"cGF$I\"aGF$I\"dGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "B:=\{a,b,d\}; A; C minus B; C minus A;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG6\"I\"aGF$I\"dGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#I\"cG6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "<$I\"cG6\"I\"dGF$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 72 "A; B:=\{b,c\}; C minus (A union B); (C minus A ) intersect (C minus B);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\" I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"cGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#I\"dG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#I \"dG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "C minus (A inter sect B); (C minus A) union (C minus B);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"cG6\"I\"aGF$I\"dGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"cG6\"I\"aGF$I\"dGF$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" " %#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.26. Feladat." } }}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.27. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.28. Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.29. Fela dat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.30. Feladat." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.2.31. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.2.32. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.2.33. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.34. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" " %#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.35. Feladat." } }{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "1.2.36. Hatv\303\241nyhalmaz." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "with(combin at,powerset): powerset(\{a,b,c\}); powerset(\{a,b\});\n" }{MPLTEXT 1 0 34 "powerset(\{a\}); powerset(\{\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<*<\"<%I\"bG6\"I\"cGF&I\"aGF&<$F%F(<$F'F(<#F%<#F'<#F(<$F%F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "<&<\"<$I\"bG6\"I\"aGF&<#F%<#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$<\"<#I\"aG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 45 "* 1.2.37. Hatv\303\241nyhalmaz-a xi\303\263ma." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.38. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.39. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.2.40. Feladat ." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.2.41. Feladat." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 51 "* 1.2.42. V\303\251gtelens\303\251gi axi\303\263ma. " }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.2.43. Megjegyz" }{TEXT 207 8 "\303\251" } {TEXT 207 3 "s. " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "1.2.44. Tov " }{TEXT 207 8 "\303\241" }{TEXT 207 15 "bbi feladatok r" }{TEXT 207 8 "\303\251" }{TEXT 207 14 "szletes megold" }{TEXT 207 8 "\303\241" } {TEXT 207 5 "ssal." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "1.2.45. T ov" }{TEXT 207 8 "\303\241" }{TEXT 207 14 "bbi feladatok." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 27 "1.3. Rel\303\241ci\303\263k" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 28 "1.3.1. \+ Rendezett p\303\241r." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 41 "A rendezett p\303\241r a Maple-ben [x,y]." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ev alb(\{\{x\},\{x,y\}\}=\{\{y\},\{x,y\}\});evalb([x,y]=[y,x]);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p:=[x,y]; p[1]; p[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$I\"xG6\"I\"yGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"xG6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"yG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`type/ordpair`:=proc(x) type(x,list) and nops(x)=2 en d;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 46 " type([a,b],ordpair); typ e([1,2,3],ordpair);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%F %F%3-I%typeG%*protectedG6$F$I%listGF)/-I%nopsGF)F#\"\"#F%F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->1.3.2. Fela dat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 25 "1.3.3. Descartes-szorzat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "bincar tprod:=proc(X::set,Y::set) local x,y,Z; Z:=\{\};\n" }{MPLTEXT 1 0 67 " for x in X do for y in Y do Z:=Z union \{[x,y]\}; od; od; Z; end;\n" } }{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"YGF& F'6%I\"xGF&I\"yGF&I\"ZGF&F&F&C%>F.<\"?&F,F%I%trueGF(?&F-F*F3>F.-I&unio nGF(6$F.<#7$F,F-F.F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "bincartprod(\{1,2,3\},\{a,b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$ \"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$F*F(7$\"\"$F%7$F-F(" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->1.3.4. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->1.3.5. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "1.3.6. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "1.3.7. Feladat." } }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 43 "1.3.8. Bin\303\251r rel\303\24 1ci\303\263k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "binrel:=proc(R::set(ordpair),X::set,Y::set) loca l r;\n" }{MPLTEXT 1 0 65 "if nargs<1 or nargs>3 then error \": needs 1 ..3 arguments\" fi;\n" }{MPLTEXT 1 0 15 "for r in R do\n" }{MPLTEXT 1 0 63 " if nargs=2 and not (member(r[1],X) and member(r[2],X)) then\n" }{MPLTEXT 1 0 19 " return false;\n" }{MPLTEXT 1 0 65 " elif nargs =3 and not (member(r[1],X) and member(r[2],Y)) then\n" }{MPLTEXT 1 0 18 " return false\n" }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 16 "od ; true; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"RG6\"-I$setG%* protectedG6#I(ordpairGF&'I\"XGF&F('I\"YGF&F(6#I\"rGF&F&F&C%@$52%&nargs G\"\"\"2\"\"$F6YQ7:~needs~1..3~argumentsF&?&F1F%I%trueGF)@&3/F6\"\"#43 -I'memberGF)6$&F16#F7F--FE6$&F16#FAF-OI&falseGF)3/F6F943FD-FE6$FKF/FMF =F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "binrel(\{\});\n" }{MPLTEXT 1 0 78 "R:=bincartprod(\{1,2,3\},\{a,b\});binrel(R);binrel( R,\{0,1,2,3\},\{a,b,c\});\n" }{MPLTEXT 1 0 54 "binrel(R,\{a,b\},\{1,2, 3\});binrel(R,\{1,2,3,a,b\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%tru eG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$ I\"aGF&7$\"\"#F%7$F*F(7$\"\"$F%7$F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protect edG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "id:=proc(X) local x; map(x->[x,x],X); end; id(\{1,3,a \});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"XG6\"6#I\"xGF%F%F%-I$ma pG%*protectedG6$f*F&F%6$I)operatorGF%I&arrowGF%F%7$F'F'F%F%F%F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$\"\"\"F$7$\"\"$F&7$I\"aG6\"F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "F:=\{\{1\},\{2\},\{1,2\}\} ; FF:=bincartprod(F,F); select(x->x[1] subset x[2],FF);select(x->x[1] \+ subset x[2] and not x[1]=x[2],FF);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " <%<#\"\"\"<#\"\"#<$F$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<+7$<#\"\"#<# \"\"\"7$F$F$7$F$<$F'F%7$F*F&7$F*F$7$F*F*7$F&F&7$F&F$7$F&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$<#\"\"#F$7$F$<$\"\"\"F%7$F'F'7$<#F(F+7$F+F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$7$<#\"\"#<$\"\"\"F%7$<#F'F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "irem(13,5); X:=\{1,2,3,4,5,6 \};XX:=bincartprod(X,X):\n" }{MPLTEXT 1 0 37 "R:=select(x->irem(x[2],x [1])=0,XX);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$\"\"\"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$\"\"'F,7$F$F(7$F $\"\"%7$F$F*7$F$F,7$F%F%7$F%F/7$F%F,7$F(F,7$F/F/" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 15 "1.3.9. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "1.3.10. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "1.3.11. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "1.3.12. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "1.3.13. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 45 "1.3. 14. Rel\303\241ci\303\263k gr\303\241fja." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 86 "1.3. 15. \303\211rtelmez\303\251si tartom\303\241ny, \303\251rt\303\251kk\3 03\251szlet." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "R:=\{[1,a],[1,b],[2,b],[3,d],[2,d],[4,e]\};\n" } {MPLTEXT 1 0 57 "dmn:=proc(R::set(ordpair)) map(x->x[1],R); end; dmn(R );\n" }{MPLTEXT 1 0 57 "rng:=proc(R::set(ordpair)) map(x->x[2],R); end ; rng(R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$I\" aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"RG6\"-I$setG%*protectedG6#I(ordpairGF&F&F&F&-I$ma pGF)6$f*6#I\"xGF&F&6$I)operatorGF&I&arrowGF&F&&F16#\"\"\"F&F&F&F%F&F&F &" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"RG6\"-I$setG%*protectedG6#I(ordpairGF&F& F&F&-I$mapGF)6$f*6#I\"xGF&F&6$I)operatorGF&I&arrowGF&F&&F16#\"\"#F&F&F &F%F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"aGF$I\"dGF$I\" eGF$" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1. 3.16. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 61 "1.3.17. Kiter jeszt\303\251s, lesz\305\261k\303\255t\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "R; select(x ->(x[1]>1 and x[2]<>b),R);\n" }{MPLTEXT 1 0 66 "restrict:=proc(R::set( ordpair),X) select(x->(x[1] in X),R); end;\n" }{MPLTEXT 1 0 28 "X:=\{2 ,3\}; restrict(R,X);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"b G6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$\"\"$I\"dG6\"7$\"\"#F%7$\"\"%I\"eGF&" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"RG6\"-I$setG%*protectedG6#I(ordp airGF&I\"XGF&F&F&F&-I'selectGF)6$f*6#I\"xGF&F&6$I)operatorGF&I&arrowGF &F&-I#inGF)6$&F26#\"\"\"F,F&F&6$F,9%F%F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$\"\"#I \"bG6\"7$\"\"$I\"dGF&7$F$F)" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 15 "1.3.18. Inverz." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "relinv:=proc(R::set(ordpair)) map(x->[x[2],x[1]],R); end;\n" } {MPLTEXT 1 0 61 "R; dmn(R); rng(R); S:=relinv(R); dmn(S); rng(S); reli nv(S);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"RG6\"-I$setG%*protec tedG6#I(ordpairGF&F&F&F&-I$mapGF)6$f*6#I\"xGF&F&6$I)operatorGF&I&arrow GF&F&7$&F16#\"\"#&F16#\"\"\"F&F&F&F%F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F -7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&\"\"\"\"\"#\"\"$\" \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"aGF$I\"dGF$I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$I\"bG6\"\"\"#7$I\"dGF%\"\"$7$F(F&7 $I\"eGF%\"\"%7$F$\"\"\"7$I\"aGF%F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "<& I\"bG6\"I\"aGF$I\"dGF$I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&\"\" \"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7 $F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 56 "1.3.19. Halmaz k\303\251pe \303\251s inverz k\303\251pe. " }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "mapset:=proc(R::set(ordpair),A::set) rng(restrict(R,A)) end; \n" }{MPLTEXT 1 0 73 "invmapset:=proc(R::set(ordpair),A::set) rng(rest rict(relinv(R),A)) end;\n" }{MPLTEXT 1 0 68 "R; S:=relinv(R); A:=\{1,4 \}; B:=\{b,e\}; mapset(R,A); mapset(R,B);\n" }{MPLTEXT 1 0 30 "invmaps et(R,B); mapset(S,B);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"RG6\" -I$setG%*protectedG6#I(ordpairGF&'I\"AGF&F(F&F&F&-I$rngGF&6#-I)restric tGF&6$F%F-F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"RG6\"-I$setG %*protectedG6#I(ordpairGF&'I\"AGF&F(F&F&F&-I$rngGF&6#-I)restrictGF&6$- I'relinvGF&6#F%F-F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\" bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$I\"bG6\"\"\"#7$I\"dGF%\"\"$7$F(F&7$I\"eGF% \"\"%7$F$\"\"\"7$I\"aGF%F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\" \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG6\"I\"aGF$I\"eGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"\"\"\"#\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"\"\"\"#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.20. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "- >1.3.21. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.22. \+ Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.23. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 33 "1.3.24. Kompoz\303\255ci\303 \263." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "relcomp:=proc(R::set(ordpair),S::set(ordpair)) local \+ r,s,T; T:=\{\};\n" }{MPLTEXT 1 0 29 "for r in R do for s in S do\n" } {MPLTEXT 1 0 53 " if s[2]=r[1] then T:=T union \{[s[1],r[2]]\}; fi;\n " }{MPLTEXT 1 0 17 "od; od; T; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 47 "R; S:=\{[aa,1],[bb,3],[cc,5]\}; relcomp(R,S);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"RG6\"-I$setG%*protectedG6#I(ordpairGF&'I\"S GF&F'6%I\"rGF&I\"sGF&I\"TGF&F&F&C%>F1<\"?&F/F%I%trueGF)?&F0F-F6@$/&F06 #\"\"#&F/6#\"\"\">F1-I&unionGF)6$F1<#7$&F0F>&F/F;F1F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"d GF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$I#aaG6\" \"\"\"7$I#bbGF%\"\"$7$I#ccGF%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<% 7$I#aaG6\"I\"bGF%7$F$I\"aGF%7$I#bbGF%I\"dGF%" }}}{EXCHG {PARA 0 "" 0 " " {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1. 3.25. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.26. Fel adat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 37 "1.3.27. \303\201ll\303 \255t\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 78 "R; S:=\{[aa,1],[aa,2],[bb,3],[cc,4],[dd,5]\}; \+ relcomp(R,S); rng(R); rng(%%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$ \"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$I#aaG6\"\"\"\"7$I#bbGF%\"\"$7$F$\" \"#7$I#ccGF%\"\"%7$I#ddGF%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$I #aaG6\"I\"bGF%7$F$I\"aGF%7$I#bbGF%I\"dGF%7$F$F+7$I#ccGF%I\"eGF%" }} {PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"aGF$I\"dGF$I\"eGF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "<&I\"bG6\"I\"aGF$I\"dGF$I\"eGF$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "T:=\{[xx,aa],[xx,cc]\}; relc omp(R,relcomp(S,T)); relcomp(relcomp(R,S),T);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$7$I#xxG6\"I#aaGF%7$F$I#ccGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&7$I#xxG6\"I\"bGF%7$F$I\"aGF%7$F$I\"dGF%7$F$I\"eGF%" }} {PARA 11 "" 1 "" {XPPMATH 20 "<&7$I#xxG6\"I\"bGF%7$F$I\"aGF%7$F$I\"dGF %7$F$I\"eGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "relinv(relc omp(R,S)); relcomp(relinv(S),relinv(R));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$I\"bG6\"I#aaGF%7$I\"aGF%F&7$I\"dGF%I#bbGF%7$F*F&7$I\" eGF%I#ccGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$I\"bG6\"I#aaGF%7$I\"a GF%F&7$I\"dGF%I#bbGF%7$F*F&7$I\"eGF%I#ccGF%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 37 "1.3 .28. \303\201ll\303\255t\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "R; IX:=id(\{1,2,3,4,5\}); r elcomp(R,IX); IY:=id(\{a,b,c,d,e\}); relcomp(IY,R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7 $F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$\"\"\"F$7$\" \"$F&7$\"\"&F(7$\"\"#F*7$\"\"%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$ \"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$I\"aG6\"F$7$I\"bGF%F'7$I\"cGF%F)7$ I\"dGF%F+7$I\"eGF%F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"I\"bG6 \"7$F$I\"aGF&7$\"\"#F%7$\"\"$I\"dGF&7$F*F-7$\"\"%I\"eGF&" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 32 "1.3.29. Defin\303\255ci\303\263." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "istransitiv e:=proc(R::set(ordpair)) local r,s;\n" }{MPLTEXT 1 0 29 "for r in R do for s in R do\n" }{MPLTEXT 1 0 46 " if r[2]=s[1] and not [r[1],s[2]] in R then\n" }{MPLTEXT 1 0 22 " return false fi;\n" }{MPLTEXT 1 0 20 "od; od; true; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 43 "X:=\{ 1,2,3,4,5,6\}; XX:=bincartprod(X,X):\n" }{MPLTEXT 1 0 54 "R:=select(x- >irem(x[2],x[1])=0,XX); istransitive(R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"RG6\"-I$setG%*protectedG6#I(ordpairGF&6$I\"rGF&I \"sGF&F&F&C$?&F-F%I%trueGF)?&F.F%F1@$3/&F-6#\"\"#&F.6#\"\"\"4-I#inGF)6 $7$&F-F:&F.F7F%OI&falseGF)F1F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<( \"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$ \"\"\"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$\"\"'F,7$F$F(7$F$\"\"%7$F$F*7$F$F ,7$F%F%7$F%F/7$F%F,7$F(F,7$F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%tru eG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "issymmetr ic:=proc(R::set(ordpair)) local r,s;\n" }{MPLTEXT 1 0 15 "for r in R d o\n" }{MPLTEXT 1 0 49 " if not [r[2],r[1]] in R then return false fi; \n" }{MPLTEXT 1 0 16 "od; true; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 17 "issymmetric(R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"RG 6\"-I$setG%*protectedG6#I(ordpairGF&6$I\"rGF&I\"sGF&F&F&C$?&F-F%I%true GF)@$4-I#inGF)6$7$&F-6#\"\"#&F-6#\"\"\"F%OI&falseGF)F1F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "isantisymmetric:=proc(R::set(ordpair)) local r;\n " }{MPLTEXT 1 0 15 "for r in R do\n" }{MPLTEXT 1 0 68 " if [r[2],r[1] ] in R then if r[1]<>r[2] then return false fi; fi;\n" }{MPLTEXT 1 0 16 "od; true; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 21 "isantisym metric(R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"RG6\"-I$setG%*pr otectedG6#I(ordpairGF&6#I\"rGF&F&F&C$?&F-F%I%trueGF)@$-I#inGF)6$7$&F-6 #\"\"#&F-6#\"\"\"F%@$0F9F6OI&falseGF)F0F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "isstriclyantisymmetric:=proc(R::set(ordpair)) local r;\n" } {MPLTEXT 1 0 15 "for r in R do\n" }{MPLTEXT 1 0 45 " if [r[2],r[1]] i n R then return false fi;\n" }{MPLTEXT 1 0 16 "od; true; end;\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 28 "isstriclyantisymmetric(R);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"RG6\"-I$setG%*protectedG6#I(ordp airGF&6#I\"rGF&F&F&C$?&F-F%I%trueGF)@$-I#inGF)6$7$&F-6#\"\"#&F-6#\"\" \"F%OI&falseGF)F0F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*pro tectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "isreflexive:=pro c(X::set,R::set(ordpair)) local x;\n" }{MPLTEXT 1 0 42 "if not binrel( R,X) then return false fi;\n" }{MPLTEXT 1 0 70 "for x in X do if not [ x,x] in R then return false fi; od; true; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 19 "isreflexive(X,R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&6#I\"xGF&F&F &C%@$4-I'binrelGF&6$F*F%OI&falseGF(?&F/F%I%trueGF(@$4-I#inGF(6$7$F/F/F *F6F9F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "isirreflexive:=proc(X::set,R ::set(ordpair)) local x;\n" }{MPLTEXT 1 0 42 "if not binrel(R,X) then \+ return false fi;\n" }{MPLTEXT 1 0 66 "for x in X do if [x,x] in R then return false fi; od; true; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 21 "isirreflexive(X,R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"X G6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&6#I\"xGF&F&F&C%@$4-I'bi nrelGF&6$F*F%OI&falseGF(?&F/F%I%trueGF(@$-I#inGF(6$7$F/F/F*F6F9F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "istrichotom:=proc(X::set,R::set(ord pair)) local x,y;\n" }{MPLTEXT 1 0 42 "if not binrel(R,X) then return \+ false fi;\n" }{MPLTEXT 1 0 29 "for x in X do for y in X do\n" } {MPLTEXT 1 0 50 " if x<>y then if ([x,y] in R and [y,x] in R) or\n" } {MPLTEXT 1 0 63 " ((not [x,y] in R) and (not [y,x] in R)) then retu rn false\n" }{MPLTEXT 1 0 11 " fi; fi;\n" }{MPLTEXT 1 0 20 "od; od; t rue; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 19 "istrichotom(X,R); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I \"RGF&-F'6#I(ordpairGF&6$I\"xGF&I\"yGF&F&F&C%@$4-I'binrelGF&6$F*F%OI&f alseGF(?&F/F%I%trueGF(?&F0F%F:@$0F/F0@$53-I#inGF(6$7$F/F0F*-FB6$7$F0F/ F*45FAFEF7F:F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protecte dG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "isdichotom:=proc(X::s et,R::set(ordpair)) local x,y;\n" }{MPLTEXT 1 0 42 "if not binrel(R,X) then return false fi;\n" }{MPLTEXT 1 0 29 "for x in X do for y in X d o\n" }{MPLTEXT 1 0 58 " if not([x,y] in R or [y,x] in R) then return \+ false fi;\n" }{MPLTEXT 1 0 20 "od; od; true; end;\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 18 "isdichotom(X,R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&6$I\"xGF&I \"yGF&F&F&C%@$4-I'binrelGF&6$F*F%OI&falseGF(?&F/F%I%trueGF(?&F0F%F:@$4 5-I#inGF(6$7$F/F0F*-F@6$7$F0F/F*F7F:F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.30. Fel adat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.31. Feladat." }}}{SECT 1 {PARA 5 " " 0 "" {TEXT 207 16 "1.3.32. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.33. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.3.34. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1 .3.35. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.3.36. Fela dat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 146 "1.3.37. Reflex\303\255 v, szimmetrikus illetve tranzit\303\255v rel\303\241ci\303\263k gr\303 \241fj\303\241nak egyszer\305\261s\303\255t\303\251se." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }} }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 69 "1.3.38. Ekvivalenciarel\303\24 1ci\303\263, oszt\303\241lyoz\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "isequivalence:=proc(X ::set,R::set(ordpair))\n" }{MPLTEXT 1 0 63 "istransitive(R) and issymm etric(R) and isreflexive(X,R); end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 21 "isequivalence(X,R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I \"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&F&F&F&33-I-istransit iveGF&6#F*-I,issymmetricGF&F2-I,isreflexiveGF&6$F%F*F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "E:=select(x->irem(x[1],3)=irem(x[2],3),XX); isequiv alence(X,E);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<.7$\"\"\"F$7$\"\"$F&7 $\"\"&F(7$\"\"'F&7$F*F*7$F$\"\"%7$\"\"#F/7$F/F(7$F&F*7$F-F$7$F-F-7$F(F /" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "ispartition:=proc(X::set,cO::set(se t)) local Y,Z;\n" }{MPLTEXT 1 0 49 "for Y in cO do if Y=\{\} then retu rn false fi;\n" }{MPLTEXT 1 0 133 " for Z in cO do if Y<>Z and Y int ersect Z<>\{\} then return false; fi; od; od; if `union`(op(cO))<>X th en false else true fi; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I \"XG6\"I$setG%*protectedG'I#cOGF&-F'6#F'6$I\"YGF&I\"ZGF&F&F&C$?&F.F*I% trueGF(C$@$/F.<\"OI&falseGF(?&F/F*F2@$30F.F/0-I*intersectGF(F-F6F7@%0- I&unionGF(6#-I#opGF(6#F*F%F8F2F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "X; cO:=\{\{1,4\},\{2,5\},\{3,6\}\}; ispartition(X,cO) ;\n" }{MPLTEXT 1 0 45 "cO:=\{\{1\},\{2,3,4\}\}; ispartition(X,cO);\n" }{MPLTEXT 1 0 51 "cO:=\{\{1,2,3\},\{4,5,6,7\}\}; ispartition(X,cO);\n" }{MPLTEXT 1 0 51 "cO:=\{\{1,2,3,4\},\{4,5,6\}\}; ispartition(X,cO);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'" } }{PARA 11 "" 1 "" {XPPMATH 20 "<%<$\"\"\"\"\"%<$\"\"#\"\"&<$\"\"$\"\"' " }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$<#\"\"\"<%\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$< &\"\"%\"\"&\"\"'\"\"(<%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$<%\"\"%\" \"&\"\"'<&\"\"\"\"\"#\"\"$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG %*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.3.39. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "eq ui2part:=proc(X::set,E::set(ordpair)) local cO,x,y,tx; cO:=\{\};\n" } {MPLTEXT 1 0 25 "for x in X do tx:=\{\};\n" }{MPLTEXT 1 0 63 " for y \+ in X do if [x,y] in E then tx:=tx union \{y\} fi; od;\n" }{MPLTEXT 1 0 23 " cO:=cO union \{tx\}\n" }{MPLTEXT 1 0 14 "od; cO; end;\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 27 "X; E; cO:=equi2part(X,E);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"EGF&- F'6#I(ordpairGF&6&I#cOGF&I\"xGF&I\"yGF&I#txGF&F&F&C%>F/<\"?&F0F%I%true GF(C%>F2F5?&F1F%F7@$-I#inGF(6$7$F0F1F*>F2-I&unionGF(6$F2<#F1>F/-FB6$F/ <#F2F/F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\"\"\"\"#\"\"$\"\"% \"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<.7$\"\"\"F$7$\"\"$F&7$\" \"&F(7$\"\"'F&7$F*F*7$F$\"\"%7$\"\"#F/7$F/F(7$F&F*7$F-F$7$F-F-7$F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%<$\"\"\"\"\"%<$\"\"#\"\"&<$\"\"$\"\" '" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "part2equi:=proc(X::set ,cO::set(set)) local E,Y,x,y; E:=\{\};\n" }{MPLTEXT 1 0 44 "for Y in c O do for x in Y do for y in Y do\n" }{MPLTEXT 1 0 24 " E:=E union \{[ x,y]\}\n" }{MPLTEXT 1 0 21 "od; od; od; E; end;\n" }{MPLTEXT 1 0 2 "\n " }{MPLTEXT 1 0 18 "part2equi(X,cO);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I#cOGF&-F'6#F'6&I\"EGF&I\"YGF&I\"xGF &I\"yGF&F&F&C%>F.<\"?&F/F*I%trueGF(?&F0F/F6?&F1F/F6>F.-I&unionGF(6$F.< #7$F0F1F.F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<.7$\"\"\"F$7$\"\"$F& 7$\"\"&F(7$\"\"'F&7$F*F*7$F$\"\"%7$\"\"#F/7$F/F(7$F&F*7$F-F$7$F-F-7$F( F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "cO; equi2part(X,part2 equi(X,cO)); E; part2equi(X,equi2part(X,E));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%<$\"\"\"\"\"%<$\"\"#\"\"&<$\"\"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%<$\"\"\"\"\"%<$\"\"#\"\"&<$\"\"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<.7$\"\"\"F$7$\"\"$F&7$\"\"&F(7$\"\"'F&7$F*F*7$F$\" \"%7$\"\"#F/7$F/F(7$F&F*7$F-F$7$F-F-7$F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "<.7$\"\"\"F$7$\"\"$F&7$\"\"&F(7$\"\"'F&7$F*F*7$F$\"\"%7$ \"\"#F/7$F/F(7$F&F*7$F-F$7$F-F-7$F(F/" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 9 "1.3.4 0. P" }{TEXT 207 12 "\303\251lda." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.41. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "- >1.3.42. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.43. \+ Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.44. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 55 "1.3.45. R\303\251szbenrendez\303\251s, rendez\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "is partialordering:=proc(X::set,R::set(ordpair))\n" }{MPLTEXT 1 0 67 "ist ransitive(R) and isantisymmetric(R) and isreflexive(X,R); end;\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 31 "X; R; ispartialordering(X,R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RG F&-F'6#I(ordpairGF&F&F&F&33-I-istransitiveGF&6#F*-I0isantisymmetricGF& F2-I,isreflexiveGF&6$F%F*F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\" \"\"\"\"#\"\"$\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$\" \"\"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$\"\"'F,7$F$F(7$F$\"\"%7$F$F*7$F$F,7 $F%F%7$F%F/7$F%F,7$F(F,7$F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG %*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "iscomparabl e:=proc(x,y,R::set(ordpair))\n" }{MPLTEXT 1 0 39 "evalb([x,y] in R or \+ [y,x] in R); end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 43 "iscomparab le(2,6,R); iscomparable(2,3,R);\n" }{MPLTEXT 1 0 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"xG6\"I\"yGF%'I\"RGF%-I$setG%*protectedG6#I(ord pairGF%F%F%F%-I&evalbGF+6#5-I#inGF+6$7$F$F&F(-F36$7$F&F$F(F%F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "isordering:=proc(X::set,R::set(ordpair)) local x;\n" }{MPLTEXT 1 0 49 "ispartialordering(X,R) and isdichotom(X,R) end;\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 64 "isordering(X,R); S:=select(x->x[ 1]<=x[2],XX); isordering(X,S);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$ 'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&6#I\"xGF&F&F&3-I2i spartialorderingGF&6$F%F*-I+isdichotomGF&F3F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "<77 $\"\"\"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$F*\"\"'7$F,F,7$F$F(7$F$\"\"%7$F$ F*7$F$F,7$F%F%7$F%F(7$F%F07$F%F*7$F%F,7$F(F07$F(F*7$F(F,7$F0F07$F0F*7$ F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ischain:=proc(X::set,R::set(ordpair )) local S;\n" }{MPLTEXT 1 0 34 "S:=R intersect bincartprod(X,X);\n" } {MPLTEXT 1 0 23 "isordering(X,S); end;\n" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 45 "ischain(\{1,2,4\},R); ischain(\{1,2,3\},R);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF&- F'6#I(ordpairGF&6#I\"SGF&F&F&C$>F/-I*intersectGF(6$F*-I,bincartprodGF& 6$F%F%-I+isorderingGF&6$F%F/F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I% trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protected G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.3.46. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 61 "1.3.47. Szigor\303\272 \303\251s gyenge rel\303\241ci\303\263" } {TEXT 207 1 "." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 56 "1.3.48. Szigor\303\272 \303\251s gyenge rendez\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "strictrel:=proc(X::set,R::set(ordpa ir)) R minus id(X); end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 26 "X; \+ R; S:=strictrel(X,R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\" I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&F&F&F&-I&minusGF(6$F*-I#idG F&6#F%F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\"\"\"\"#\"\"$\"\"% \"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$\"\"\"\"\"#7$F$F$7$\" \"$F(7$\"\"&F*7$\"\"'F,7$F$F(7$F$\"\"%7$F$F*7$F$F,7$F%F%7$F%F/7$F%F,7$ F(F,7$F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "<*7$\"\"\"\"\"#7$F$\"\"$7$ F$\"\"%7$F$\"\"&7$F$\"\"'7$F%F)7$F%F-7$F'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "weakrel:=proc(X::set,R::set(ordpair)) R union id(X ); end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 15 "weakrel(X,R);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF&- F'6#I(ordpairGF&F&F&F&-I&unionGF(6$F*-I#idGF&6#F%F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$\"\"\"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$\"\"'F,7$F $F(7$F$\"\"%7$F$F*7$F$F,7$F%F%7$F%F/7$F%F,7$F(F,7$F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "istransitive(S); isirreflexive(X,S) ; isstriclyantisymmetric(S);\n" }{MPLTEXT 1 0 17 "istrichotom(X,S);" } }{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%t rueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "R:=select(x->x[1]<=x[2] ,XX); S:=strictrel(X,R); istrichotom(X,S);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<77$\"\"\"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$F*\"\"'7$F,F,7$F $F(7$F$\"\"%7$F$F*7$F$F,7$F%F%7$F%F(7$F%F07$F%F*7$F%F,7$F(F07$F(F*7$F( F,7$F0F07$F0F*7$F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "<17$\"\"\"\"\"#7 $\"\"&\"\"'7$F$\"\"$7$F$\"\"%7$F$F'7$F$F(7$F%F*7$F%F,7$F%F'7$F%F(7$F*F ,7$F*F'7$F*F(7$F,F'7$F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*pr otectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.49. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.50. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 22 "1.3.51. Intervallumok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "int_o_o:=proc(X::set,R::s et(ordpair),x,y) local S,z; S:=\{\};\n" }{MPLTEXT 1 0 16 "for z in X d o \n" }{MPLTEXT 1 0 77 " if [x,z] in R and x<>z and [z,y] in R and z< >y then S:=S union \{z\}; fi;\n" }{MPLTEXT 1 0 13 "od; S; end;\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 63 "int_o_c:=proc(X::set,R::set(ordp air),x,y) local S,z; S:=\{\};\n" }{MPLTEXT 1 0 16 "for z in X do \n" } {MPLTEXT 1 0 68 " if [x,z] in R and x<>z and [z,y] in R then S:=S uni on \{z\}; fi;\n" }{MPLTEXT 1 0 13 "od; S; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 61 "int_i_o:=proc(X::set,R::set(ordpair),x) local S,y; \+ S:=\{\};\n" }{MPLTEXT 1 0 16 "for y in X do \n" }{MPLTEXT 1 0 53 " if [y,x] in R and x<>y then S:=S union \{y\}; fi;\n" }{MPLTEXT 1 0 13 "o d; S; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&'I\"XG6\"I$setG%*pro tectedG'I\"RGF&-F'6#I(ordpairGF&I\"xGF&I\"yGF&6$I\"SGF&I\"zGF&F&F&C%>F 1<\"?&F2F%I%trueGF(@$333-I#inGF(6$7$F.F2F*0F.F2-F=6$7$F2F/F*0F2F/>F1-I &unionGF(6$F1<#F2F1F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&'I\"XG6 \"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&I\"xGF&I\"yGF&6$I\"SGF&I \"zGF&F&F&C%>F1<\"?&F2F%I%trueGF(@$33-I#inGF(6$7$F.F2F*0F.F2-F<6$7$F2F /F*>F1-I&unionGF(6$F1<#F2F1F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6 %'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&I\"xGF&6$I\"SGF&I \"yGF&F&F&C%>F0<\"?&F1F%I%trueGF(@$3-I#inGF(6$7$F1F.F*0F.F1>F0-I&union GF(6$F0<#F1F0F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "X:= \{1,2,3,4,5,6\}; XX:=bincartprod(X,X):\n" }{MPLTEXT 1 0 37 "R:=select( x->irem(x[2],x[1])=0,XX);\n" }{MPLTEXT 1 0 53 "int_o_o(X,R,1,6); int_o _c(X,R,1,6); int_i_o(X,R,6);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\" \"\"\"#\"\"$\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$\"\" \"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$\"\"'F,7$F$F(7$F$\"\"%7$F$F*7$F$F,7$F %F%7$F%F/7$F%F,7$F(F,7$F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"#\" \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"#\"\"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "$ 2..6;$ 1..9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\" \"#\"\"$\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+\"\"\"\"\"# \"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 57 "1.3. 52. R\303\251szbenrendez\303\251sek Hasse-diagrammja." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 80 "1.3.53. Legkisebb, legnagyobb, minim\303\241lis \303\251s maxi m\303\241lis elem." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "least:=proc(X::set,R::set(ordpair)) local x ,y,f;\n" }{MPLTEXT 1 0 25 "for x in X do f:=true; \n" }{MPLTEXT 1 0 65 " for y in X do if not [x,y] in R then f:=false; break; fi; od;\n" }{MPLTEXT 1 0 26 " if f then return x fi;\n" }{MPLTEXT 1 0 16 "od; N ULL; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 53 "greatest:=proc(X:: set,R::set(ordpair)) local x,y,f;\n" }{MPLTEXT 1 0 25 "for x in X do f :=true; \n" }{MPLTEXT 1 0 65 " for y in X do if not [y,x] in R then f :=false; break; fi; od;\n" }{MPLTEXT 1 0 26 " if f then return x fi; \n" }{MPLTEXT 1 0 16 "od; NULL; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&6%I\"xGF&I\" yGF&I\"fGF&F&F&C$?&F/F%I%trueGF(C%>F1F4?&F0F%F4@$4-I#inGF(6$7$F/F0F*C$ >F1I&falseGF([@$F1OF/I%NULLGF(F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 " f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&6%I\"xGF&I\"yG F&I\"fGF&F&F&C$?&F/F%I%trueGF(C%>F1F4?&F0F%F4@$4-I#inGF(6$7$F0F/F*C$>F 1I&falseGF([@$F1OF/I%NULLGF(F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "X; R; least(X,R); greatest(X,R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$\"\"\"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$\"\"'F,7$F$F(7$F $\"\"%7$F$F*7$F$F,7$F%F%7$F%F/7$F%F,7$F(F,7$F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "mins :=proc(X::set,R::set(ordpair)) local x,y,f,S; S:=\{\};\n" }{MPLTEXT 1 0 25 "for x in X do f:=true; \n" }{MPLTEXT 1 0 70 " for y in X do if \+ [y,x] in R and x<>y then f:=false; break; fi; od;\n" }{MPLTEXT 1 0 35 " if f then S:=S union \{x\}; fi;\n" }{MPLTEXT 1 0 13 "od; S; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 60 "maxs:=proc(X::set,R::set(ordpa ir)) local x,y,f,S; S:=\{\};\n" }{MPLTEXT 1 0 25 "for x in X do f:=tru e; \n" }{MPLTEXT 1 0 70 " for y in X do if [x,y] in R and x<>y then f :=false; break; fi; od;\n" }{MPLTEXT 1 0 35 " if f then S:=S union \{ x\}; fi;\n" }{MPLTEXT 1 0 13 "od; S; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&6 &I\"xGF&I\"yGF&I\"fGF&I\"SGF&F&F&C%>F2<\"?&F/F%I%trueGF(C%>F1F7?&F0F%F 7@$3-I#inGF(6$7$F0F/F*0F/F0C$>F1I&falseGF([@$F1>F2-I&unionGF(6$F2<#F/F 2F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protecte dG'I\"RGF&-F'6#I(ordpairGF&6&I\"xGF&I\"yGF&I\"fGF&I\"SGF&F&F&C%>F2<\"? &F/F%I%trueGF(C%>F1F7?&F0F%F7@$3-I#inGF(6$7$F/F0F*0F/F0C$>F1I&falseGF( [@$F1>F2-I&unionGF(6$F2<#F/F2F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "X; R; mins(X,R); maxs(X,R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$\"\"\"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$\"\"'F,7$F$F(7$F $\"\"%7$F$F*7$F$F,7$F%F%7$F%F/7$F%F,7$F(F,7$F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"%\"\"&\" \"'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.3.54. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 24 "1.3.55. Korl\303\241tok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "islowerbound:=proc(X::set,R: :set(ordpair),Y::set,x) local y;\n" }{MPLTEXT 1 0 70 "for y in Y do if not [x,y] in R then return false fi; od; true; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 62 "isupperbound:=proc(X::set,R::set(ordpair),Y:: set,x) local y;\n" }{MPLTEXT 1 0 70 "for y in Y do if not [y,x] in R t hen return false fi; od; true; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&'I\"YGF&F'I\" xGF&6#I\"yGF&F&F&C$?&F2F/I%trueGF(@$4-I#inGF(6$7$F0F2F*OI&falseGF(F5F& F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&'I\"XG6\"I$setG%*protectedG' I\"RGF&-F'6#I(ordpairGF&'I\"YGF&F'I\"xGF&6#I\"yGF&F&F&C$?&F2F/I%trueGF (@$4-I#inGF(6$7$F2F0F*OI&falseGF(F5F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "X; R; islowerbound(X,R,\{2,3,5\},1); isupperbound(X, R,\{2,3,5\},6);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\"\"\"\"#\"\"$ \"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$\"\"\"\"\"#7$F$F$ 7$\"\"$F(7$\"\"&F*7$\"\"'F,7$F$F(7$F$\"\"%7$F$F*7$F$F,7$F%F%7$F%F/7$F% F,7$F(F,7$F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" } }{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "lowerbounds:=proc(X::set,R::set(ordpair),Y ::set) local S,x; S:=\{\}; \n" }{MPLTEXT 1 0 71 "for x in X do if islo werbound(X,R,Y,x) then S:=S union \{x\} fi; od;\n" }{MPLTEXT 1 0 9 "S ; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"XG6\"I$setG%*protect edG'I\"RGF&-F'6#I(ordpairGF&'I\"YGF&F'6$I\"SGF&I\"xGF&F&F&C%>F1<\"?&F2 F%I%trueGF(@$-I-islowerboundGF&6&F%F*F/F2>F1-I&unionGF(6$F1<#F2F1F&F&F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "upperbounds:=proc(X::s et,R::set(ordpair),Y::set) local S,x; S:=\{\};\n" }{MPLTEXT 1 0 71 "fo r x in X do if isupperbound(X,R,Y,x) then S:=S union \{x\} fi; od;\n" }{MPLTEXT 1 0 9 "S; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"X G6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&'I\"YGF&F'6$I\"SGF&I\"x GF&F&F&C%>F1<\"?&F2F%I%trueGF(@$-I-isupperboundGF&6&F%F*F/F2>F1-I&unio nGF(6$F1<#F2F1F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "X:= \{1,2,3,4,5,6\}; XX:=bincartprod(X,X):\n" }{MPLTEXT 1 0 37 "R:=select( x->irem(x[2],x[1])=0,XX);\n" }{MPLTEXT 1 0 53 "lowerbounds(X,R,\{2,4\} ); upperbounds(X,R,\{2,4\});\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\" \"\"\"#\"\"$\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<07$\"\" \"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$\"\"'F,7$F$F(7$F$\"\"%7$F$F*7$F$F,7$F %F%7$F%F/7$F%F,7$F(F,7$F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\" \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "inf:=proc(X::set,R::set(ordpair),Y::set)\n" } {MPLTEXT 1 0 38 "greatest(lowerbounds(X,R,Y),R); end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 42 "sup:=proc(X::set,R::set(ordpair),Y::set)\n" }{MPLTEXT 1 0 35 "least(upperbounds(X,R,Y),R); end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 37 "inf(X,R,\{2,4\}); sup(X,R,\{2,4\});\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"XG6\"I$setG%*protectedG'I\"RGF&- F'6#I(ordpairGF&'I\"YGF&F'F&F&F&-I)greatestGF&6$-I,lowerboundsGF&6%F%F *F/F*F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"XG6\"I$setG%*prot ectedG'I\"RGF&-F'6#I(ordpairGF&'I\"YGF&F'F&F&F&-I&leastGF&6$-I,upperbo undsGF&6%F%F*F/F*F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1. 3.56. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.57. Fel adat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.58. Feladat." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.59. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.60. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.61. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1 .3.62. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.63. Fe ladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 30 "->1.3.64. Feladat: program fut" }{TEXT 207 8 "\303\241" }{TEXT 207 9 "si sebess" }{TEXT 207 8 "\303\251" }{TEXT 207 1 "g" } {TEXT 207 8 "\303\251" }{TEXT 207 13 "nek optimaliz" }{TEXT 207 8 "\30 3\241" }{TEXT 207 1 "l" }{TEXT 207 8 "\303\241" }{TEXT 207 4 "sa r" } {TEXT 207 8 "\303\251" }{TEXT 207 11 "szbenrendez" }{TEXT 207 8 "\303 \251" }{TEXT 207 12 "s kiterjeszt" }{TEXT 207 8 "\303\251" }{TEXT 207 10 "seinek seg" }{TEXT 207 8 "\303\255" }{TEXT 207 2 "ts" }{TEXT 207 8 "\303\251" }{TEXT 207 1 "g" }{TEXT 207 8 "\303\251" }{TEXT 207 4 "ve l." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 34 "1.3.65. J\303\263lrendez\303\251s." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "iswellordering:=proc(X::set,R::set(ordpair)) local f,Y,P; \n" } {MPLTEXT 1 0 48 "f:=isordering(X,R); if not f then return f fi;\n" } {MPLTEXT 1 0 17 "P:=powerset(X);\n" }{MPLTEXT 1 0 15 "for Y in P do\n" }{MPLTEXT 1 0 67 " if Y<>\{\} then f:=least(Y,R); if f=NULL then ret urn f; fi; fi;\n" }{MPLTEXT 1 0 16 "od; true; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairG F&6%I\"fGF&I\"YGF&I\"PGF&F&F&C'>F/-I+isorderingGF&6$F%F*@$4F/OF/>F1-_I )combinatG6$F(I(_syslibGF&I)powersetGF&6#F%?&F0F1I%trueGF(@$0F0<\"C$>F /-I&leastGF&6$F0F*@$/F/I%NULLGF(F9FCF&F&F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "X; R; iswellordering(X,R);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 50 "R:=select(x->x[1]<=x[2],XX);iswellordering(X,R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "<07$\"\"\"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7 $\"\"'F,7$F$F(7$F$\"\"%7$F$F*7$F$F,7$F%F%7$F%F/7$F%F,7$F(F,7$F/F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "<77$\"\"\"\"\"#7$F$F$7$\"\"$F(7$\"\"&F*7$F*\"\"'7$F,F,7$ F$F(7$F$\"\"%7$F$F*7$F$F,7$F%F%7$F%F(7$F%F07$F%F*7$F%F,7$F(F07$F(F*7$F (F,7$F0F07$F0F*7$F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protec tedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.3.66. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.67. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3 .68. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.69. Fela dat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.70. Feladat." }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "1.3.71. P\303\251ld\303\241k." } }{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "orderingprod:=proc(X::set,R::set(ordpair),Y::set,S::set(ordpair) )\n" }{MPLTEXT 1 0 32 "local x,y,xp,yp,RS; RS:=\{\}; \n" }{MPLTEXT 1 0 60 "for x in X do for y in Y do for xp in X do for yp in Y do \n" } {MPLTEXT 1 0 77 " if [x,xp] in R and [y,yp] in S then RS:=RS union \{ [[x,y],[xp,yp]]\}; fi;\n" }{MPLTEXT 1 0 26 "od; od; od; od; RS; end;\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&'I\"XG6\"I$setG%*protectedG'I\"R GF&-F'6#I(ordpairGF&'I\"YGF&F''I\"SGF&F+6'I\"xGF&I\"yGF&I#xpGF&I#ypGF& I#RSGF&F&F&C%>F7<\"?&F3F%I%trueGF(?&F4F/FF7-I&unionGF(6$F7<#7$7$F3F47$F5F6F7F&F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "X:=\{1,2\}; R:=\{[1,1],[1,2] ,[2,2]\}; isordering(X,R);\n" }{MPLTEXT 1 0 56 "Y:=\{a,b\}; S:=\{[a,a] ,[a,b],[b,b]\}; isordering(Y,S);\n" }{MPLTEXT 1 0 50 "XY:=bincartprod( X,Y); RS:=orderingprod(X,R,Y,S);\n" }{MPLTEXT 1 0 20 "isordering(XY,RS );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "<%7$\"\"\"\"\"#7$F$F$7$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I \"bG6\"I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$I\"aG6\"F$7$I\"bGF %F'7$F$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }} {PARA 11 "" 1 "" {XPPMATH 20 "<&7$\"\"\"I\"bG6\"7$F$I\"aGF&7$\"\"#F%7$ F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "<+7$7$\"\"\"I\"bG6\"F$7$F$7$\"\" #F&7$7$F%I\"aGF'F$7$F,F,7$F,F)7$F,7$F*F-7$F)F)7$F1F)7$F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 103 "strictorderingprod:=proc(X::set,R::set(ordpair),Y: :set,S::set(ordpair)) local x,y,xp,yp,RS; RS:=\{\};\n" }{MPLTEXT 1 0 60 "for x in X do for y in Y do for xp in X do for yp in Y do \n" } {MPLTEXT 1 0 60 " if [x,xp] in R and x<>xp and [y,yp] in S and y<>yp \+ then \n" }{MPLTEXT 1 0 43 " RS:=RS union \{[[x,y],[xp,yp]]\}; fi;\n " }{MPLTEXT 1 0 26 "od; od; od; od; RS; end;\n" }{MPLTEXT 1 0 2 "\n" } }{PARA 11 "" 1 "" {XPPMATH 20 "f*6&'I\"XG6\"I$setG%*protectedG'I\"RGF& -F'6#I(ordpairGF&'I\"YGF&F''I\"SGF&F+6'I\"xGF&I\"yGF&I#xpGF&I#ypGF&I#R SGF&F&F&C%>F7<\"?&F3F%I%trueGF(?&F4F/FF7-I&unionGF(6$F7<#7$7$F3F47$F5F6F7F&F &F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "lexorderingprod:=pro c(X::set,R::set(ordpair),Y::set,S::set(ordpair))\n" }{MPLTEXT 1 0 32 " local x,y,xp,yp,RS; RS:=\{\}; \n" }{MPLTEXT 1 0 60 "for x in X do for \+ y in Y do for xp in X do for yp in Y do \n" }{MPLTEXT 1 0 62 " if ([x ,xp] in R and x<>xp) or (x=xp and [y,yp] in S) then \n" }{MPLTEXT 1 0 43 " RS:=RS union \{[[x,y],[xp,yp]]\}; fi;\n" }{MPLTEXT 1 0 26 "od; od; od; od; RS; end;\n" }{MPLTEXT 1 0 2 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&'I\"XG6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&' I\"YGF&F''I\"SGF&F+6'I\"xGF&I\"yGF&I#xpGF&I#ypGF&I#RSGF&F&F&C%>F7<\"?& F3F%I%trueGF(?&F4F/FF7-I&unionGF(6$F7<#7$7$F3F47$F5F6F7F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "strictrel(XY,RS); strictorderingpro d(X,R,Y,S);\n" }{MPLTEXT 1 0 41 "strictrel(XY,lexorderingprod(X,R,Y,S) );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$7$\"\"\"I\"bG6\"7$\"\"#F&7$7 $F%I\"aGF'F$7$F+F(7$F+7$F)F,7$F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "<# 7$7$\"\"\"I\"aG6\"7$\"\"#I\"bGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7$ 7$\"\"\"I\"bG6\"7$\"\"#F&7$7$F%I\"aGF'F$7$F+F(7$F+7$F)F,7$F/F(7$F$F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sort([cc,ca,cb,bb,aa,ab,b a],lexorder);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7)I#aaG6\"I#abGF$I#baGF $I#bbGF$I#caGF$I#cbGF$I#ccGF$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.72. Feladat ." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.3.73. Feladat." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.3.74. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.3.75. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.3.76. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "1.2.77. Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 15 "bbi felada tok r" }{TEXT 207 8 "\303\251" }{TEXT 207 14 "szletes megold" }{TEXT 207 8 "\303\241" }{TEXT 207 5 "ssal." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "1.2.78. Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 14 "bbi felad atok." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 29 "1.4. F\303\274ggv\303\251nyek" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 30 "1.4.1. F\303\274ggv\303\251ny." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "isfunction:=proc(f::set(ordpair),X::set,Y::set) local x,y,S;\n" }{MPLTEXT 1 0 43 "if not binrel(args) then return false fi;\n" } {MPLTEXT 1 0 48 "for x in dmn(f) do S:=\{\}; for y in rng(f) do\n" } {MPLTEXT 1 0 43 " if [x,y] in f then S:=S union \{y\} fi;\n" } {MPLTEXT 1 0 42 " od; if nops(S)>1 then return false fi;\n" }{MPLTEXT 1 0 16 "od; true; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"fG6 \"-I$setG%*protectedG6#I(ordpairGF&'I\"XGF&F('I\"YGF&F(6%I\"xGF&I\"yGF &I\"SGF&F&F&C%@$4-I'binrelGF&6#%%argsGOI&falseGF)?&F1-I$dmnGF&6#F%I%tr ueGF)C%>F3<\"?&F2-I$rngGF&F@FA@$-I#inGF)6$7$F1F2F%>F3-I&unionGF)6$F3<# F2@$2\"\"\"-I%nopsGF)6#F3F;FAF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f:=id(\{a,b\}); isfunction(f); isfunction(f,\{a,b,1\} );\n" }{MPLTEXT 1 0 72 "isfunction(f,\{a,b\},\{1,2\});f:=\{[a,1],[b,2] ,[a,2]\}; isfunction(f);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$7$I\"aG6 \"F$7$I\"bGF%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 " " {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "< %7$I\"aG6\"\"\"#7$I\"bGF%F&7$F$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "is injective:=proc(f::set(ordpair))\n" }{MPLTEXT 1 0 47 "isfunction(f) an d isfunction(relinv(f)); end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 61 "isinjective(\{[a,1],[b,2]\}); isinjective(\{[a,1],[b,1]\});\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"fG6\"-I$setG%*protectedG6#I(ordp airGF&F&F&F&3-I+isfunctionGF&6#F%-F.6#-I'relinvGF&F/F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "issurjective:=proc(f::set(ordpair),Y::set)\n" } {MPLTEXT 1 0 34 "isfunction(f) and rng(f)=Y; end;\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 40 "issurjective(\{[a,1],[b,1]\},\{1,2\});\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 38 "issurjective(\{[a,1],[b,2]\},\{1 ,2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"fG6\"-I$setG%*protecte dG6#I(ordpairGF&'I\"YGF&F(F&F&F&3-I+isfunctionGF&6#F%/-I$rngGF&F1F-F&F &F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "isbijective:=proc(f::set(ordpair),Y::set)\n" } {MPLTEXT 1 0 44 "isinjective(f) and issurjective(f,Y); end;\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 39 "isbijective(id(\{1,2,3\}),\{1,2, 3\});\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 39 "isbijective(\{[a,1],[b ,2]\},\{1,2,3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"fG6\"-I$set G%*protectedG6#I(ordpairGF&'I\"YGF&F(F&F&F&3-I,isinjectiveGF&6#F%-I-is surjectiveGF&6$F%F-F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*pr otectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 " " "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f:=x->x^2; f(1) ;f(2);f(3); eval(f); type(f,procedure);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF%I&arrowGF%F%*$)F$\"\"#\"\"\" F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF%I&arrowGF%F%*$)F$\" \"#\"\"\"F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f:='f'; eval(f); whattype (f);\n" }{MPLTEXT 1 0 19 "f(1):=1; eval(f);\n" }{MPLTEXT 1 0 19 "f(2): =4; f(3):=8;\n" }{MPLTEXT 1 0 20 "f(1);f(2);f(3);f(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"fG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"fG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I'symbolG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6\"F#6#I)re memberGF#E\\s\"\"\"\"F'.-%)procnameG6#%%argsGF#F#F#" }}{PARA 11 "" 1 " " {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"fG6\"6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "isarr owfromto:=proc(f::procedure,X::set,Y::set) local x;\n" }{MPLTEXT 1 0 69 "for x in X do if not f(x) in Y then return false fi; od; true; end ;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 42 "isarrowfromto(f,\{1,2,3\}, \{1,4,8,10\});\n" }{MPLTEXT 1 0 33 "isarrowfromto(f,\{1,2\},\{1,8\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"fG6\"I*procedureG%*protectedG 'I\"XGF&I$setGF('I\"YGF&F+6#I\"xGF&F&F&C$?&F/F*I%trueGF(@$4-I#inGF(6$- F%F.F-OI&falseGF(F2F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*pr otectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "makefunction:=proc(R::set(or dpair)) local x,y,f;\n" }{MPLTEXT 1 0 43 "if not isfunction(R) then re turn NULL fi;\n" }{MPLTEXT 1 0 70 "for x in dmn(R) do for y in rng(R) \+ do if [x,y] in R then f(x):=y fi;\n" }{MPLTEXT 1 0 23 "od; od; eval(f) ; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 56 "f:='f'; R:=\{[1,1],[2 ,4],[3,9]\}; f(1);f(2);f(3);f(4);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 41 "f:=makefunction(R);f(1);f(2);f(3);f(4);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"RG6\"-I$setG%*protectedG6#I(ordpairGF&6%I\"xGF&I \"yGF&I\"fGF&F&F&C%@$4-I+isfunctionGF&6#F%OI%NULLGF)?&F--I$dmnGF&F5I%t rueGF)?&F.-I$rngGF&F5F;@$-I#inGF)6$7$F-F.F%>-F/6#F-F.-I%evalGF)6#F/F&F &F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"fG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$\"\"\"F$7$\"\"#\"\"%7$\"\"$\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"fG6\"6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"fG 6\"6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"fG6\"6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"fG6\"6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6\"F#6#I)rememberGF#E\\s$\"\"\"F'\"\"#\"\"%\"\"$\"\"*.-%)procnam eG6#%%argsGF#F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I\"fG6\"6#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->1.4.2. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->1.4.3. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->1.4.4. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->1.4.5. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "1.4.6. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->1.4.7. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "1.4.8. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "1.4.9. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.4.10. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 37 "1.4.11. \303\201ll\303\255t\303\241s." }}{EXCHG {PARA 0 "" 0 " " {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 42 "1. 4.12. Kanonikus lek\303\251pez\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "makecanonical:=proc(X ::set,R::set(ordpair)) local x,rx,y,f;\n" }{MPLTEXT 1 0 48 "if not ise quivalence(X,R) then return FAIL fi;\n" }{MPLTEXT 1 0 39 "for x in X d o rx:=\{\}; for y in X do\n" }{MPLTEXT 1 0 58 " if [x,y] in R then rx :=rx union \{y\} fi; od; f(x):=rx\n" }{MPLTEXT 1 0 18 "od; eval(f) end ;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 64 "f:=makecanonical(\{1,2,3\} ,\{[1,1],[1,2],[2,1],[2,2],[3,3]\});\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 17 "f(1);f(2);f(3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG 6\"I$setG%*protectedG'I\"RGF&-F'6#I(ordpairGF&6&I\"xGF&I#rxGF&I\"yGF&I \"fGF&F&F&C%@$4-I.isequivalenceGF&6$F%F*OI%FAILGF(?&F/F%I%trueGF(C%>F0 <\"?&F1F%F<@$-I#inGF(6$7$F/F1F*>F0-I&unionGF(6$F0<#F1>-F26#F/F0-I%eval GF(6#F2F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6\"F#6#I)rememberGF#E \\s$\"\"\"<$F'\"\"#F)F(\"\"$<#F*.-%)procnameG6#%%argsGF#F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.13. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.4.14. Feladat ." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.4.15. Feladat." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.4.16. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*1.4.17. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 41 "1.4.18. Monoton f\303\274ggv\303\251nyek." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{PARA 0 "" 0 "" {TEXT 201 107 "Az al \303\241bbi k\303\251t f\303\274ggv\303\251nyben R rendez\303\251s X-e n, S pedig rendez\303\251s Y-on." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "isincreasing:=proc(f::proced ure,X::set,R::set(ordpair),Y::set,S::set(ordpair)) local x,y;\n" } {MPLTEXT 1 0 50 "if not isarrowfromto(f,X,Y) then return FAIL fi;\n" } {MPLTEXT 1 0 73 "if not ispartialordering(X,R) or not ispartialorderin g(Y,S) then return\n" }{MPLTEXT 1 0 12 " FAIL fi;\n" }{MPLTEXT 1 0 29 "for x in X do for y in X do\n" }{MPLTEXT 1 0 64 " if [x,y] in R a nd not [f(x),f(y)] in S then return false fi;\n" }{MPLTEXT 1 0 19 "od; od; true end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 73 "X:=\{1,2,3\}; XX:=bincartprod(X,X): R:=select(x->irem(x[2],x[1])=0,XX);\n" } {MPLTEXT 1 0 94 "Y:=X; S:=select(x->x[1]<=x[2],XX); f:=x->x; isincreas ing(f,X,R,Y,S);isincreasing(f,Y,S,X,R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6''I\"fG6\"I*procedureG%*protectedG'I\"XGF&I$setGF('I\" RGF&-F+6#I(ordpairGF&'I\"YGF&F+'I\"SGF&F.6$I\"xGF&I\"yGF&F&F&C&@$4-I.i sarrowfromtoGF&6%F%F*F2OI%FAILGF(@$43-I2ispartialorderingGF&6$F*F--FD6 $F2F4F>?&F6F*I%trueGF(?&F7F*FI@$3-I#inGF(6$7$F6F7F-4-FN6$7$-F%6#F6-F%6 #F7F4OI&falseGF(FIF&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"\"\"\" #\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7$\"\"\"\"\"#7$F$F$7$\"\"$F( 7$F$F(7$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"\"\"\"#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "<(7$\"\"\"\"\"#7$F$F$7$\"\"$F(7$F$F(7$F% F%7$F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF %I&arrowGF%F%F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protec tedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.19. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.20. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" " %#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 33 "1.4.21. Indexelt csal \303\241dok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "issetfamily:=proc(Iset::set,f::procedure) local i ;\n" }{MPLTEXT 1 0 62 "for i in Iset do if not type(f(i),set) then ret urn false fi;\n" }{MPLTEXT 1 0 16 "od; true; end;\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 64 "f:='f'; f(1):=\{a,b\};f(2):=\{b,c,d\}; issetfami ly(\{1,2\},f);\n" }{MPLTEXT 1 0 27 "issetfamily(\{1,2,3\},f);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I%IsetG6\"I$setG%*protectedG'I\"fGF &I*procedureGF(6#I\"iGF&F&F&C$?&F-F%I%trueGF(@$4-I%typeGF(6$-F*F,F'OI& falseGF(F0F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"fG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I\"bG6\"I\"aGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG6\"I\"cGF$I\"dGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG% *protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "1.4.22. De Morgan-szab\303\241lyok." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "1.4.2 3. Megjegyz\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "`union`();" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "`inte rsect`();" }}{PARA 8 "" 1 "" {TEXT 208 77 "Error, invalid input: `inte rsect` expects 1 or more arguments, but received 0" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "1.4.24. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "% #%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.4.25. Feladat." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.26. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.4.27. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.4.28. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 9 "1.4.29. R" }{TEXT 207 32 "el\303\241ci\303\263 \303\251s " } {TEXT 207 18 "Descartes-szorzat " }{TEXT 207 8 "\303\241" }{TEXT 207 4 "ltal" }{TEXT 207 8 "\303\241" }{TEXT 207 12 "nos esetben." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "s :=x,y; s[1]; s[2]; t:=y,x; evalb(s=t);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$I\"xG6\"I\"yGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"xG6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"yG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$I\"yG6\"I\"xGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protect edG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "descartesprod:=proc( L::list(set)) local x,y,i,S,SS;\n" }{MPLTEXT 1 0 56 "if nops(L)=0 then return \{\} fi; S:=map(x->[x],L[1]);\n" }{MPLTEXT 1 0 69 "for i from \+ 2 to nops(L) do SS:=\{\}; for x in S do for y in L[i] do\n" }{MPLTEXT 1 0 30 " SS:=SS union \{[op(x),y]\}\n" }{MPLTEXT 1 0 27 "od; od; S:=S S od; S; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 20 "descartesprod([]);\n" }{MPLTEXT 1 0 27 "descartesprod([\{1,2\} ]);\n" }{MPLTEXT 1 0 35 "descartesprod([\{1,2\},\{a,b\}]);\n" } {MPLTEXT 1 0 32 "descartesprod([\{1,2\},\{\}]);\n" }{MPLTEXT 1 0 41 "d escartesprod([\{1,2\},\{a\},\{x,y\}]);\n" }{MPLTEXT 1 0 45 "descartesp rod([\{1,2\},\{a,b,c\},\{x,y\}]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f *6#'I\"LG6\"-I%listG%*protectedG6#I$setGF)6'I\"xGF&I\"yGF&I\"iGF&I\"SG F&I#SSGF&F&F&C&@$/-I%nopsGF)6#F%\"\"!O<\">F0-I$mapGF)6$f*6#F-F&6$I)ope ratorGF&I&arrowGF&F&7#F-F&F&F&&F%6#\"\"\"?(F/\"\"#FGF5I%trueGF)C%>F1F: ?&F-F0FJ?&F.&F%6#F/FJ>F1-I&unionGF)6$F1<#7$-I#opGF)F@F.>F0F1F0F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$7#\"\"\"7#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&7$\"\"\"I\"bG6\" 7$F$I\"aGF&7$\"\"#F%7$F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }} {PARA 11 "" 1 "" {XPPMATH 20 "<&7%\"\"\"I\"aG6\"I\"xGF&7%F$F%I\"yGF&7% \"\"#F%F'7%F+F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "<.7%\"\"\"I\"aG6\"I \"xGF&7%F$F%I\"yGF&7%\"\"#F%F'7%F+F%F)7%F$I\"bGF&F'7%F$F.F)7%F+F.F'7%F +F.F)7%F$I\"cGF&F'7%F$F3F)7%F+F3F'7%F+F3F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "isselection:=proc(Iset::set,f::procedure,x::procedu re) local i;\n" }{MPLTEXT 1 0 49 "if not issetfamily(Iset,f) then retu rn FAIL fi;\n" }{MPLTEXT 1 0 74 "for i in Iset do if not x(i) in f(i) \+ then return false fi; od; true end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 69 "Iset:=\{a,b,c\}; f:='f'; f(a):=\{1,2\};f(b):=\{1,3\};f(c):=\{2 ,3\};\n" }{MPLTEXT 1 0 49 "x(a):=1;x(b):=3;x(c):=3; isselection(Iset,f ,x);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 31 "x(b):=2; isselection(Is et,f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I%IsetG6\"I$setG%*prote ctedG'I\"fGF&I*procedureGF('I\"xGF&F+6#I\"iGF&F&F&C%@$4-I,issetfamilyG F&6$F%F*OI%FAILGF(?&F/F%I%trueGF(@$4-I#inGF(6$-F-F.-F*F.OI&falseGF(F9F &F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG6\"I\"cGF$I\"aGF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"fG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%* protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "agent:= \{[D209,\"Peti\"],[KISZ1,\"Fleto\"],[Puf3,\"Gyula\"]\};\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 212 "event:=\{[KISZ1,\"Balaton\",19930706],[P uf3,\"Nyugati\",19561108],[KISZ1,\"Motim\",19961231],[D209,\"Paks\",20 000103],[KISZ1,\"Fittelina\",19980320],[D209,\"Gresham\",20010908],[KI SZ1,\"Nomentana\",19951122]\};\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 139 "descartesprod([agent,event]):select(x->x[1][1]=x[2][1],%):map(x-> [x[1][2],x[2][2],x[2][3],x[2][1]],%):active:=select(x->x[3]>19891023,% );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7$I%D209G6\"Q%PetiF%7$I&KISZ1G F%Q&FletoF%7$I%Puf3GF%Q&GyulaF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<)7%I &KISZ1G6\"Q(BalatonF%\")12$*>7%I%Puf3GF%Q(NyugatiF%\")36c>7%F$Q&MotimF %\")J7'*>7%I%D209GF%Q%PaksF%\").,+?7%F$Q*FittelinaF%\")?.)*>7%F0Q(Gres hamF%\")34,?7%F$Q*NomentanaF%\")A6&*>" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(7&Q%Peti6\"Q%PaksF%\").,+?I%D209GF%7&F$Q(GreshamF%\")34,?F(7&Q&Flet oF%Q(BalatonF%\")12$*>I&KISZ1GF%7&F-Q&MotimF%\")J7'*>F07&F-Q*Fittelina F%\")?.)*>F07&F-Q*NomentanaF%\")A6&*>F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "isselection:=proc(Iset::set,f::procedure,x::procedure ) local i;\n" }{MPLTEXT 1 0 49 "if not issetfamily(Iset,f) then return FAIL fi;\n" }{MPLTEXT 1 0 74 "for i in Iset do if not x(i) in f(i) th en return false fi; od; true end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 69 "Iset:=\{a,b,c\}; f:='f'; f(a):=\{1,2\};f(b):=\{1,3\};f(c):=\{2,3 \};\n" }{MPLTEXT 1 0 49 "x(a):=1;x(b):=3;x(c):=3; isselection(Iset,f,x );\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 31 "x(b):=2; isselection(Iset ,f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I%IsetG6\"I$setG%*protect edG'I\"fGF&I*procedureGF('I\"xGF&F+6#I\"iGF&F&F&C%@$4-I,issetfamilyGF& 6$F%F*OI%FAILGF(?&F/F%I%trueGF(@$4-I#inGF(6$-F-F.-F*F.OI&falseGF(F9F&F &F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"bG6\"I\"cGF$I\"aGF$" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"fG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%* protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.30. Fel adat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.31. Feladat." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.4.32. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.33. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" " %#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.34. Feladat." } }{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.35. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 25 "1.4. 36. M\305\261veletek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "isbinop:=proc(X::set,f::procedure) \+ local x,y;\n" }{MPLTEXT 1 0 29 "for x in X do for y in X do\n" } {MPLTEXT 1 0 44 " if not f(x,y) in X then return false fi;\n" } {MPLTEXT 1 0 19 "od; od; true end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 70 "f:='f'; f(0,0):=0;f(0,1):=1;f(1,0):=1;f(1,1):=0; isbinop(\{0,1 \},f);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 21 "isbinop(\{0,1,2\},f); " }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"f GF&I*procedureGF(6$I\"xGF&I\"yGF&F&F&C$?&F-F%I%trueGF(?&F.F%F1@$4-I#in GF(6$-F*F,F%OI&falseGF(F1F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"fG 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG %*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "&+(0,0):=0;&+(0,1):=1;&+(1, 0):=1;&+(1,1):=0;\n" }{MPLTEXT 1 0 20 "0&+1;1&+1;&+(0,1);\n" }{MPLTEXT 1 0 33 "isbinop(\{0,1\},(x,y)->x &+ y);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "isunop:=proc(X::set,f::procedure) local x;\n" }{MPLTEXT 1 0 68 "for x in X do if not f(x) in X then return false fi; od; true end; \n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 66 "f:='f'; f(0):=1;f(1):=0; is unop(\{0,1\},f); isunop(\{0,1,2\},f);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"XG6\"I$setG%*protectedG'I\"fGF&I*procedureGF(6#I\"xGF&F&F& C$?&F-F%I%trueGF(@$4-I#inGF(6$-F*F,F%OI&falseGF(F0F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "I\"fG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protec tedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "&inc(0):=1;&inc(1): =0; &inc 0; &inc 1; isunop(\{0,1\},x->&inc x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "9();9(x);9(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "isnullop:=proc(X::set,f::procedure) evalb(f() in X) end;\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 68 "f:='f'; f():=1; eval(f); isnullo p(\{0,1\},f); isnullop(\{0,2\},f);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " f*6$'I\"XG6\"I$setG%*protectedG'I\"fGF&I*procedureGF(F&F&F&-I&evalbGF( 6#-I#inGF(6$-F*F&F%F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"fG6\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6\"F#6#I)rememberGF#E\\s\"F#\"\"\".-%)procnameG6#%%argsGF#F#F#" }} {PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.4.37. P\303 \251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "1.4.38. P\303\251ld\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "X:=\{1,2,3\}; P:= combinat[powerset](X); isbinop(P,(x,y)->x union y);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "< *<\"<#\"\"\"<#\"\"#<$F%F'<#\"\"$<$F%F*<%F%F'F*<$F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 87 "1.4. 39. P\303\251ld\303\241k: m\305\261velet megad\303\241sa t\303\241bl\3 03\241zattal." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "true and true; true and false; false and true; f alse and false;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 13 "T:=table(); \n" }{MPLTEXT 1 0 42 "T[true,true]:=true;T[true,false]:=false;\n" } {MPLTEXT 1 0 44 "T[false,true]:=true;T[false,false]:=false;\n" } {MPLTEXT 1 0 9 "print(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*pr otectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }} {PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6 \"I&falseG%*protectedGE\\[l!" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG% *protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }} {PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I& TABLEG%*protectedG6#7&/6$I&falseGF$F)F)/6$I%trueGF$F)F)/6$F,F,F,/6$F)F ,F," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.40. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->1.4.41. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.4.42. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 53 "1.4.43. M\305\261veletek f\303\274ggv\3 03\251nyekkel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "f:=x->x^2; g:=x->x^3; (f*g)(2); (f*g)(3); (f/g) (2); (f/g)(0); (g/f)(0);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6 \"F%6$I)operatorGF%I&arrowGF%F%*$)F$\"\"#\"\"\"F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF%I&arrowGF%F%*$)F$\"\"$\" \"\"F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$V#" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"\"\"\"\"#" }} {PARA 8 "" 1 "" {TEXT 208 42 "Error, numeric exception: division by ze ro" }}{PARA 8 "" 1 "" {TEXT 208 42 "Error, numeric exception: division by zero" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.4.44. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "1.4.45. P\303\251ld\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=[true,true,false,fals e];\n" }{MPLTEXT 1 0 29 "g:=[true,false,true,false];\n" }{MPLTEXT 1 0 26 "zip((x,y)->x and y,f,g);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&I%tr ueG%*protectedGF#I&falseGF$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&I%tru eG%*protectedGI&falseGF$F#F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&I%true G%*protectedGI&falseGF$F%F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 61 "1.4.46. M\305\261ve lettart\303\263 lek\303\251pez\303\251sek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "ishom:=proc(phi:: procedure,X::set,f::procedure,Y::set,g::procedure)\n" }{MPLTEXT 1 0 12 "local x,y;\n" }{MPLTEXT 1 0 52 "if not isarrowfromto(phi,X,Y) then return FAIL fi;\n" }{MPLTEXT 1 0 42 "if not isbinop(X,f) then return \+ FAIL fi;\n" }{MPLTEXT 1 0 42 "if not isbinop(Y,g) then return FAIL fi; \n" }{MPLTEXT 1 0 29 "for x in X do for y in X do\n" }{MPLTEXT 1 0 58 " if phi(f(x,y))<>g(phi(x),phi(y)) then return false fi;\n" }{MPLTEXT 1 0 19 "od; od; true end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 72 "X :=\{true,false\}; Y:=X; ishom(x->x,X,(x,y)->x and y,Y,(x,y)->x or y); \n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 52 "ishom(x-> not x,X,(x,y)->x \+ and y,Y,(x,y)->x or y);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6''I$phiG 6\"I*procedureG%*protectedG'I\"XGF&I$setGF('I\"fGF&F''I\"YGF&F+'I\"gGF &F'6$I\"xGF&I\"yGF&F&F&C'@$4-I.isarrowfromtoGF&6%F%F*F/OI%FAILGF(@$4-I (isbinopGF&6$F*F-F;@$4-F@6$F/F1F;?&F3F*I%trueGF(?&F4F*FG@$0-F%6#-F-F2- F16$-F%6#F3-F%6#F4OI&falseGF(FGF&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I&falseG%*protectedGI%trueGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$I& falseG%*protectedGI%trueGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG% *protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "1.4.47. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a:='a'; a^(x+y); exp and(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"aG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ")I\"aG6\",&I\"xGF$\"\"\"I\"yGF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&)I\"aG6\"I\"xGF%\"\"\")F$I\"yGF%F'" }}}{EXCHG {PARA 0 " " 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "1.4.48. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "1.4.49. Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 14 "bbi feladatok." }}}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 205 35 "2. Term\303\251szetes sz\303\241 mok" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 53 "3. A sz\303\241mfogalom b\305\221v\303\2 55t\303\251se" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 205 24 "4. V\303\251ges halmazok" }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 27 "5. V\303\251gtelen halmazok" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 28 "6 . Sz\303\241melm\303\251let" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 28 "7. Gr\303\241felm\30 3\251let" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 10 "8. Algebra" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 24 "9. K \303\263dol\303\241s" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 16 "10. Algoritmusok" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }} }} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }