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Halmazok" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 35 "2. Term\303\251s zetes sz\303\241mok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 205 53 "3. A sz\303\241mfogalom b\305\221v\303\2 55t\303\251se" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 31 "3.1. Eg\303\251sz sz\303\241mok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 74 "3.1.1. Oszt\303\241lyoz\303\24 1s kompatibilit\303\241sa m\305\261velettel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "`type/ordpair`:=p roc(x) type(x,list) and nops(x)=2 end;" }}{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%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "iscompbinop:=p roc(X::set,E::set(ordpair),f::procedure) local x,xx,y,yy;\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 33 " for xx in X do f or yy in X do\n" }{MPLTEXT 1 0 67 " if [x,xx] in E and [y,yy] in E \+ and not [f(x,y),f(xx,yy)] in E\n" }{MPLTEXT 1 0 29 " then return \+ false fi;\n" }{MPLTEXT 1 0 28 "od; od; od; od; true; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 144 "X:=\{0,1,2,3,4,5\}; E:=\{[0,0],[0,3],[3 ,0],[3,3],[1,1],[1,4],[4,1],[4,4],[2,2],[2,5],[5,2],[5,5]\}; f:=(x,y)- >irem(x+y,6); iscompbinop(X,E,f);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f *6%'I\"XG6\"I$setG%*protectedG'I\"EGF&-F'6#I(ordpairGF&'I\"fGF&I*proce dureGF(6&I\"xGF&I#xxGF&I\"yGF&I#yyGF&F&F&C$?&F2F%I%trueGF(?&F4F%F8C$@$ 4-I#inGF(6$-F/6$F2F4F%OI&falseGF(?&F3F%F8?&F5F%F8@$33-F>6$7$F2F3F*-F>6 $7$F4F5F*4-F>6$7$F@-F/6$F3F5F*FBF8F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(\"\"!\"\"\"\"\"#\"\"$\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<.7$\"\"!F$7$F$\"\"$7$F&F$7$F&F&7$\"\"\"F*7$F*\"\"%7$F,F*7$F,F,7$\"\" #F07$F0\"\"&7$F2F07$F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\" I\"yGF%F%6$I)operatorGF%I&arrowGF%F%-I%iremG%*protectedG6$,&F$\"\"\"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 8 "3.1.2. P" }{TEXT 207 11 "\303\251lda" }{TEXT 207 1 "." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 15 "3.1.3. Feladat." }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 81 "3.1.4. Oszt\303\241lyoz\303\241s kompatibilit\303\241sa rel\303\241ci\303\263val." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "iscomprel:= proc(X::set,E::set(ordpair),R::set(ordpair)) local x,xx,y,yy;\n" } {MPLTEXT 1 0 29 "for x in X do for y in X do\n" }{MPLTEXT 1 0 33 " fo r xx in X do for yy in X do\n" }{MPLTEXT 1 0 72 " if [x,xx] in E an d [y,yy] in E and [x,y] in R and not [xx,yy] in R\n" }{MPLTEXT 1 0 29 " then return false fi;\n" }{MPLTEXT 1 0 28 "od; od; od; od; true ; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 73 "X:=\{0,1,2,3\}; E:=\{ [0,0],[0,2],[2,0],[2,2],[1,1],[1,3],[3,1],[3,3]\};\n" }{MPLTEXT 1 0 49 "R:=\{[0,1],[0,3],[2,1],[2,3]\}; iscomprel(X,E,R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"XG6\"I$setG%*protectedG'I\"EGF&-F'6#I(ordpair GF&'I\"RGF&F+6&I\"xGF&I#xxGF&I\"yGF&I#yyGF&F&F&C$?&F1F%I%trueGF(?&F3F% F7?&F2F%F7?&F4F%F7@$333-I#inGF(6$7$F1F2F*-F@6$7$F3F4F*-F@6$7$F1F3F/4-F @6$7$F2F4F/OI&falseGF(F7F&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(F&7$F*F$7$F&F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&7$\"\"!\"\"$7$F$\"\"\"7$\"\"#F'7$F)F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 8 "3.1.5 . 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T" }{TEXT 207 36 "\303\251tel: \342\204\225 be\303\241" }{TEXT 207 4 "gyaz" }{TEXT 207 27 "\303\241sa \342\204\244-be." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 10 "3.1.8. Az " }{TEXT 207 41 "eg\303\251sz \+ sz\303\241mok rendez\303\251" }{TEXT 207 3 "se." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 10 "3.1.9. Az " }{TEXT 207 40 "eg\303\251sz sz\303\241 mok szorz\303\241" }{TEXT 207 3 "sa." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 11 "3.1.10. Az " }{TEXT 207 37 "eg\303\251sz sz\303\241mok sz\303 \241" }{TEXT 207 1 "m" }{TEXT 207 8 "\303\255" }{TEXT 207 1 "t" }{TEXT 207 8 "\303\263" }{TEXT 207 1 "g" }{TEXT 207 30 "\303\251pes \303\241 br\303\241" }{TEXT 207 3 "zol" }{TEXT 207 8 "\303\241" }{TEXT 207 3 "s a." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 64 "3.1.11. Hatv\303\241nyoz \303\241s eg\303\251sz kitev\305\221vel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "(a^(-1))^5; a^(m+n); \+ expand(%); (a*b)^5;\n" }{MPLTEXT 1 0 43 "(a^n)^m; combine(%,power) ass uming integer;" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$)I\"aG6\"\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ")I\"aG6\",&I\"mGF$\"\"\"I\"nGF$F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "*&)I\"aG6\"I\"mGF%\"\"\")F$I\"nGF%F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "*&)I\"aG6\"\"\"&\"\"\")I\"bGF%F&F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "))I\"aG6\"I\"nGF%I\"mGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ")I\"aG6\"*&I\"nGF$\"\"\"I\"mGF$F'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 9 "3.1.12. P" }{TEXT 207 11 "\303\251lda" }{TEXT 207 1 "." }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "3.1.13. Gy\305\261r\305\261k." } }{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "isgrupoid:=proc(G::set,f::procedure) local x,y;\n" }{MPLTEXT 1 0 70 "for x in G do for y in G do if not f(x,y) in G then return false fi;\n" }{MPLTEXT 1 0 20 "od; od; true; end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"GG6\"I$setG%*protectedG'I\"fGF&I*procedureGF(6$I \"xGF&I\"yGF&F&F&C$?&F-F%I%trueGF(?&F.F%F1@$4-I#inGF(6$-F*F,F%OI&false GF(F1F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "neutral:=pro c(G::set,f::procedure) local x,y,s,S;\n" }{MPLTEXT 1 0 44 "if not isgr upoid(G,f) then return NULL fi;\n" }{MPLTEXT 1 0 38 "for x in G do s:= true; for y in G do\n" }{MPLTEXT 1 0 55 " if f(x,y)<>y or f(y,x)<>y t hen s:=false; break; fi;\n" }{MPLTEXT 1 0 42 "od; if s then return x f i; od; NULL end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 55 "G:=\{a,b,c \};neutral(G,(x,y)->y);neutral(G,(x,y)->y);\n" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 40 "neutral(\{0,1,2\},(x,y)->irem(x+y,3));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"GG6\"I$setG%*protectedG'I\"fGF&I*procedur eGF(6&I\"xGF&I\"yGF&I\"sGF&I\"SGF&F&F&C%@$4-I*isgrupoidGF&6$F%F*OI%NUL LGF(?&F-F%I%trueGF(C%>F/F:?&F.F%F:@$50-F*6$F-F.F.0-F*6$F.F-F.C$>F/I&fa lseGF([@$F/OF-F8F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"aG6\"I\"b GF$I\"cGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 53 "issemigroup:=proc(G::set,f::procedure) local x,y,z;\n" }{MPLTEXT 1 0 45 "if not isgrupoid(G,f) then return false f i;\n" }{MPLTEXT 1 0 43 "for x in G do for y in G do for z in G do\n" } {MPLTEXT 1 0 53 " if f(x,f(y,z))<>f(f(x,y),z) then return false fi;\n " }{MPLTEXT 1 0 23 "od; od; od; true end;\n" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 34 "issemigroup(\{a,b,c\},(x,y)->x);\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 50 "issemigroup(\{true,false\},(x,y)-> x implies y); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"GG6\"I$setG%*protectedG'I \"fGF&I*procedureGF(6%I\"xGF&I\"yGF&I\"zGF&F&F&C%@$4-I*isgrupoidGF&6$F %F*OI&falseGF(?&F-F%I%trueGF(?&F.F%F9?&F/F%F9@$0-F*6$F--F*6$F.F/-F*6$- F*6$F-F.F/F6F9F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protect edG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "isgroup:=proc(G::set,f::procedure) \+ local x,y,n,i;\n" }{MPLTEXT 1 0 45 "if not isgrupoid(G,f) then return \+ false fi;\n" }{MPLTEXT 1 0 47 "if not issemigroup(G,f) then return fal se fi;\n" }{MPLTEXT 1 0 50 "n:=neutral(G,f); if n=NULL then return fal se fi;\n" }{MPLTEXT 1 0 39 "for x in G do i:=false; for y in G do\n" } {MPLTEXT 1 0 52 " if f(x,y)=n and f(y,x)=n then i:=true; break fi;\n" }{MPLTEXT 1 0 53 "od; if i=false then return false fi; od; true; end; \n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 40 "isgroup(\{0,1,2\},(x,y)->ir em(x+y,3));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"GG6\"I$setG%*pr otectedG'I\"fGF&I*procedureGF(6&I\"xGF&I\"yGF&I\"nGF&I\"iGF&F&F&C(@$4- I*isgrupoidGF&6$F%F*OI&falseGF(@$4-I,issemigroupGF&F6F7>F/-I(neutralGF &F6@$/F/I%NULLGF(F7?&F-F%I%trueGF(C%>F0F8?&F.F%FD@$3/-F*6$F-F.F//-F*6$ F.F-F/C$>F0FD[@$/F0F8F7FDF&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%tru eG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "iscommuta tive:=proc(G::set,f::procedure) local x,y;\n" }{MPLTEXT 1 0 45 "if not isgrupoid(G,f) then return false fi;\n" }{MPLTEXT 1 0 29 "for x in G \+ do for y in G do\n" }{MPLTEXT 1 0 43 " if f(x,y)<>f(y,x) then return \+ false fi;\n" }{MPLTEXT 1 0 20 "od; od; true; end;\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 46 "iscommutative(\{0,1,2\},(x,y)->irem(x+y,3));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"GG6\"I$setG%*protectedG'I\"fGF &I*procedureGF(6$I\"xGF&I\"yGF&F&F&C%@$4-I*isgrupoidGF&6$F%F*OI&falseG F(?&F-F%I%trueGF(?&F.F%F8@$0-F*F,-F*6$F.F-F5F8F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "isabeliangroup:=proc(G::set,f::procedure)\n" } {MPLTEXT 1 0 42 "isgroup(G,f) and iscommutative(G,f) end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 46 "iscommutative(\{0,1,2\},(x,y)->irem(x+y, 3));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"GG6\"I$setG%*protected G'I\"fGF&I*procedureGF(F&F&F&3-I(isgroupGF&6$F%F*-I.iscommutativeGF&F/ F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "isleftdistributive:=proc(R:: set,f::procedure,g::procedure) local x,y,z;\n" }{MPLTEXT 1 0 45 "if no t isgrupoid(R,f) then return false fi;\n" }{MPLTEXT 1 0 45 "if not isg rupoid(R,g) then return false fi;\n" }{MPLTEXT 1 0 43 "for x in R do f or y in R do for z in R do\n" }{MPLTEXT 1 0 58 " if g(x,f(y,z))<>f(g( x,y),g(x,z)) then return false fi;\n" }{MPLTEXT 1 0 21 "od; od; od; tr ue end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"RG6\"I$setG%*protecte dG'I\"fGF&I*procedureGF('I\"gGF&F+6%I\"xGF&I\"yGF&I\"zGF&F&F&C&@$4-I*i sgrupoidGF&6$F%F*OI&falseGF(@$4-F66$F%F-F8?&F/F%I%trueGF(?&F0F%F??&F1F %F?@$0-F-6$F/-F*6$F0F1-F*6$-F-6$F/F0-F-6$F/F1F8F?F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "isrightdistributive:=proc(R::set,f: :procedure,g::procedure) local x,y,z;\n" }{MPLTEXT 1 0 45 "if not isgr upoid(R,f) then return false fi;\n" }{MPLTEXT 1 0 45 "if not isgrupoid (R,g) then return false fi;\n" }{MPLTEXT 1 0 43 "for x in R do for y i n R do for z in R do\n" }{MPLTEXT 1 0 58 " if g(f(y,z),x)<>f(g(y,x),g (z,x)) then return false fi;\n" }{MPLTEXT 1 0 21 "od; od; od; true end ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"RG6\"I$setG%*protectedG'I\" fGF&I*procedureGF('I\"gGF&F+6%I\"xGF&I\"yGF&I\"zGF&F&F&C&@$4-I*isgrupo idGF&6$F%F*OI&falseGF(@$4-F66$F%F-F8?&F/F%I%trueGF(?&F0F%F??&F1F%F?@$0 -F-6$-F*6$F0F1F/-F*6$-F-6$F0F/-F-6$F1F/F8F?F&F&F&" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "isring:=proc(R::set,f::procedure,g::procedur e)\n" }{MPLTEXT 1 0 42 "isabeliangroup(R,f) and issemigroup(R,g)\n" } {MPLTEXT 1 0 65 "and isleftdistributive(R,f,g) and isrightdistributive (R,f,g) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"RG6\"I$setG%*pro tectedG'I\"fGF&I*procedureGF('I\"gGF&F+F&F&F&333-I/isabeliangroupGF&6$ F%F*-I,issemigroupGF&6$F%F--I3isleftdistributiveGF&6%F%F*F--I4isrightd istributiveGF&F9F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "i scommutativering:=proc(R::set,f::procedure,g::procedure)\n" }{MPLTEXT 1 0 41 "isring(R,f,g) and iscommutative(R,g) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"RG6\"I$setG%*protectedG'I\"fGF&I*procedureGF('I\" gGF&F+F&F&F&3-I'isringGF&6%F%F*F--I.iscommutativeGF&6$F%F-F&F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "isringwithunity:=proc(R::set ,f::procedure,g::procedure)\n" }{MPLTEXT 1 0 41 "isring(R,f,g) and neu tral(R,g)<>NULL end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"RG6\"I$s etG%*protectedG'I\"fGF&I*procedureGF('I\"gGF&F+F&F&F&3-I'isringGF&6%F% F*F-0-I(neutralGF&6$F%F-I%NULLGF(F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "isringwithunity(\{0\},(x,y)->0,(x,y)->0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "X:=\{a,b,c\}; P:=combinat[powerset](X);\n" }{MPLTEXT 1 0 43 "isring(P,(x,y)->symmdiff(x,y),(x,y)->\{\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"aG6\"I\"bGF$I\"cGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<*<\"<%I\"aG6\"I\"bGF&I\"cGF&<$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 21 "3.1.14. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "iscommutativering(P,(x,y)->symmdiff (x,y),(x,y)->x intersect y);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 63 "isringwithunity(P,(x,y)->symmdiff(x,y),(x,y)->x intersect y);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->3.1.15. Fela dat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 29 "3.1.16. P\303\251ld\303 \241k." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->3.1.17. Feladat." }}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 18 "->3.1.18. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->3.1.19. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 73 "3.1.20. Az \303\241ltal\303\241nos disztributivit\303\241s t\3 03\251tele." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A:=sum(a[i],i=1..4); B:=sum(b[j],j=1..5); A*B; exp and(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*&I\"aG6\"6#\"\"\"F'&F$6#\" \"#F'&F$6#\"\"$F'&F$6#\"\"%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,&I\"b G6\"6#\"\"\"F'&F$6#\"\"#F'&F$6#\"\"$F'&F$6#\"\"%F'&F$6#\"\"&F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "*&,*&I\"aG6\"6#\"\"\"F(&F%6#\"\"#F(&F%6# \"\"$F(&F%6#\"\"%F(F(,,&I\"bGF&F'F(&F4F*F(&F4F-F(&F4F0F(&F46#\"\"&F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 ",J*&&I\"aG6\"6#\"\"\"F(&I\"bGF&F'F(F (*&F$F(&F*6#\"\"#F(F(*&F$F(&F*6#\"\"$F(F(*&F$F(&F*6#\"\"%F(F(*&F$F(&F* 6#\"\"&F(F(*&&F%F-F(F)F(F(*&F3.1.24. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->3.1.25. Feladat." }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 36 "3.2. Racion\30 3\241lis sz\303\241mok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 20 "3.2.1. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "type(5/7,ra tional); type(0,rational);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*p rotectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "`&~`:=(x,y)->x[1]*y[2]=y[1]* x[2];\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 47 "`&+`:=(x,y)->[x[1]*y[2 ]+x[2]*y[1],x[2]*y[2]];\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 37 "`&*` :=(x,y)->[x[1]*y[1],x[2]*y[2]];\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 51 "`&le`:=(x,y)->(y[1]*x[2]-y[2]*x[1])*x[2]*y[2]>=0;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 59 "[1,5]&~[2,10]; [1,2]&+[2,3]; [1,2]&*[2,3]; \+ [1,2]&le[2,3];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F %6$I)operatorGF%I&arrowGF%F%/*&&F$6#\"\"\"F.&F&6#\"\"#F.*&&F&F-F.&F$F0 F.F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F%6$I)ope ratorGF%I&arrowGF%F%7$,&*&&F$6#\"\"\"F/&F&6#\"\"#F/F/*&&F&F.F/&F$F1F/F /*&F5F/F0F/F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF% F%6$I)operatorGF%I&arrowGF%F%7$*&&F$6#\"\"\"F.&F&F-F.*&&F$6#\"\"#F.&F& F2F.F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F%6$I)o peratorGF%I&arrowGF%F%1\"\"!*(,&*&&F&6#\"\"\"F1&F$6#\"\"#F1F1*&&F$F0F1 &F&F3F1!\"\"F1F2F1F7F1F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "/\"#5F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"(\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "1\"\"!\"\"' " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 7 "3.2.2. " }{TEXT 207 37 "T\303\251tel: \342\204 \244 be\303\241" }{TEXT 207 4 "gyaz" }{TEXT 207 27 "\303\241sa \342\20 4\232-ba." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 39 "3.2.3. Ferdetest, test, rendezett test ." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "isskewfield:=proc(R::set,f::procedure,g::procedure) l ocal n;\n" }{MPLTEXT 1 0 50 "n:=neutral(R,f); if n=NULL then return fa lse fi;\n" }{MPLTEXT 1 0 47 "isring(R,f,g) and isgroup(R minus \{n\},g ) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"RG6\"I$setG%*protected G'I\"fGF&I*procedureGF('I\"gGF&F+6#I\"nGF&F&F&C%>F/-I(neutralGF&6$F%F* @$/F/I%NULLGF(OI&falseGF(3-I'isringGF&6%F%F*F--I(isgroupGF&6$-I&minusG F(6$F%<#F/F-F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "isfie ld:=proc(R::set,f::procedure,g::procedure) local n;\n" }{MPLTEXT 1 0 50 "n:=neutral(R,f); if n=NULL then return false fi;\n" }{MPLTEXT 1 0 54 "isring(R,f,g) and isabeliangroup(R minus \{n\},g) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"RG6\"I$setG%*protectedG'I\"fGF&I*procedu reGF('I\"gGF&F+6#I\"nGF&F&F&C%>F/-I(neutralGF&6$F%F*@$/F/I%NULLGF(OI&f alseGF(3-I'isringGF&6%F%F*F--I/isabeliangroupGF&6$-I&minusGF(6$F%<#F/F -F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "&+(0,0):=0; &+(0 ,1):=1; &+(1,0):=1; &+(1,1):=0;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 49 "&*(0,0):=0; &*(0,1):=0; &*(1,0):=0; &*(1,1):=1;\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 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "isfield(\{0,1\},(x,y)->x&+y,(x,y)->x&*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 28 "3.2.4. P\303\2 51ld\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`&+`:=(x,y)->irem(x+y,5); `&*`:=(x,y)->irem(x*y ,5); 3&+4; 3&*4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F %6$I)operatorGF%I&arrowGF%F%-I%iremG%*protectedG6$,&F$\"\"\"F&F/\"\"&F %F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F%6$I)operat orGF%I&arrowGF%F%-I%iremG%*protectedG6$*&F$\"\"\"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 7 "3.2.5 T" }{TEXT 207 36 "\303\251tel: \342 \204\232 be\303\241" }{TEXT 207 4 "gyaz" }{TEXT 207 8 "\303\241" } {TEXT 207 20 "sa rendezett testbe." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->3.2.6. Fela dat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->3.2.7. Feladat." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->3.2.8. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "3.2.9. Feladat." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 31 "3 .3. Val\303\263s sz\303\241mok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 36 "3.3.1. \303\201 ll\303\255t\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "i:='i': x:=0:\n" }{MPLTEXT 1 0 71 " for i from 0 do while (x+1)^2<2*10^(2*i) do x:=x+1: od; x; x:=x*10: od ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$S\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$T\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"%59" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"%99" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&ST\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"'?99" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"'@99" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"(5UT \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"(8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\")I@99" }}{PARA 11 "" 1 "" {XPPMATH 20 "\")N@99" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"*]8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"*c8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"+gN@99" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"+iN@99" }}{PARA 11 "" 1 "" {XPPMATH 20 "\",?c8UT \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\",Bc8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"-IiN@99" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"-PiN@99" }} {PARA 11 "" 1 "" {XPPMATH 20 "\".qBc8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\".tBc8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"/IPiN@99" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"/IPiN@99" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"0+tBc8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"04tBc8UT \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"1!4tBc8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"1&4tBc8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"2]4tBc8U T\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"2]4tBc8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"3+&4tBc8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"3/&4tB c8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"4S]4tBc8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"4[]4tBc8UT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"5! 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Komplex \+ sz\303\241mok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "`&+`:=proc(z,w) [z[1]+w[1],z[2]+w[2]] end;\n" } {MPLTEXT 1 0 64 "`&*`:=proc(z,w) [z[1]*w[1]-z[2]*w[2],z[1]*w[2]+z[2]*w [1]] end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 45 "[x,y]&+[0,0]; [x,y ]&+[-x,-y]; [x,y]&*[1,0];\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 49 "[x ,y]&*[x/(x^2+y^2),-y/(x^2+y^2)]; simplify(%);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 15 "[0,1]&*[0,1];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6 $I\"zG6\"I\"wGF%F%F%F%7$,&&F$6#\"\"\"F+&F&F*F+,&&F$6#\"\"#F+&F&F/F+F%F %F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"zG6\"I\"wGF%F%F%F%7$,&*&&F $6#\"\"\"F,&F&F+F,F,*&&F$6#\"\"#F,&F&F0F,!\"\",&*&F*F,F2F,F,*&F/F,F-F, F,F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$I\"xG6\"I\"yGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$I \"xG6\"I\"yGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&*&)I\"xG6\"\"\"#\" \"\",&*$F%F)F)*$)I\"yGF'F(F)F)!\"\"F)*&F-F)F*F/F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$!\" \"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Complex(3,5); z: =3+5*I; w:=-2-6*I; z*w; Re(z); Im(z); conjugate(z);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"$\"\"\"*&\"\"&F$^#F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"$\"\"\"*&\"\"&F$^#F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"#!\"\"*&\"\"'\"\"\"^#F'F'F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"#C\"\"\"*&\"#GF$^#F$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"$\"\"\"*&\"\"&F$^#F$F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "z:='z';w:='w'; conjugate(z);\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 66 "conjugate(conjugate(z));conjugat e(z+w);expand(%);conjugate(1/z);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "I \"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"wG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*conjugateG%*protectedG6#I\"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*conjugateG%* protectedG6#,&I\"zG6\"\"\"\"I\"wGF(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&-I*conjugateG%*protectedG6#I\"zG6\"\"\"\"-F$6#I\"wGF(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$-I*conjugateG%*protectedG6#I\"zG6\"!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 20 "3.4.2. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "64/(3^(1/2)+I); evalc (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"#k\"\"\",&*$)\"\"$#F%\"\"# F%F%^#F%F%!\"\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#;\"\"\")\"\" $#F%\"\"#F%F%*&F$F%^#F%F%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 64 "3.4.3. Komplex s z\303\241m abszol\303\272t \303\251rt\303\251ke." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "z:=x+I*y;ab s(z);evalc(%);evalc(1/(x+I*y));evalc(conjugate(z)/abs(z)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"*&^#F%F%I\"yGF$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I$absG%*protectedG6#,&I\"xG6\"\"\"\"*&^#F)F)I \"yGF(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$),&*$)I\"xG6\"\"\"#\"\" \"F**$)I\"yGF(F)F*F*#F*F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&I\"xG 6\"\"\"\",&*$)F$\"\"#F&F&*$)I\"yGF%F*F&F&!\"\"F&*(,$^#F&F&F&F-F&F'F.F. " }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&I\"xG6\"\"\"\",&*$)F$\"\"#F&F&*$ )I\"yGF%F*F&F&!\"\"F&*(,$^#F&F&F&F-F&F'F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "signum(3+4*I); signum(-5); signum(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"$\"\"&\"\"\"*&#\"\"%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 17 "->3.4.4. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->3.4.5. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->3.4.6. Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "evalc(2/(1-I)/(3+I)); evalc(1/(3+4* I)^2); evalc((2+I)/I/(-3+4*I));\n" }{MPLTEXT 1 0 52 "evalc((3^(1/2)+I) /(1-I)/(3^(1/2)-I)); simplify(%);\n" }{MPLTEXT 1 0 57 "evalc(1/I/(3-2* I)/(1+I)); evalc(1/(1-I)/(1-2*I)/(1+2*I));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"#\"\"&\"\"\"*&#F&F%F&^#F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"(\"$D'!\"\"*&#\"#CF%\"\"\"^#F*F*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"#6\"#D!\"\"*&#\"\"#F%\"\"\"^#F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**(#\"\"\"\"\"%F%,&*&#F%\"\"#F%)\"\"$F)F%F%F)! \"\"F%F+F%F%*&#F%\"\")F%F+F%F-F/F-*&^#F%F%,(*(F$F%,&F(F%F)F%F%F+F%F%F. F%F/F-F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&#\"\"\"\"\"%F%)\"\"$#F %\"\"#F%!\"\",&F$F%*&F$F%^#F%F%F%F%*&F-F%F'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"\"\"#E!\"\"*&#\"\"&F%F$^#F$F$F&" }}{PARA 11 "" 1 " " {XPPMATH 20 ",&#\"\"\"\"#5F$*&F#F$^#F$F$F$" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->3.4.7. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 15 "3.4.8. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 15 "3.4.9. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 73 "3.4.10. Komplex sz\303\241m argumentuma \303\251s tr igonometrikus alakja." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "polar(x+I*y); op(1,%); op(2,%%); po lar(3+4*I); evalc(%); argument(3+I*4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&polarGI(_syslibG6\"6$-I$absG%*protectedG6#,&I\"xGF%\"\"\"*&^#F-F- I\"yGF%F-F--I)argumentG6$F)F$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I$ab sG%*protectedG6#,&I\"xG6\"\"\"\"*&^#F)F)I\"yGF(F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "-I)argumentG6$%*protectedGI(_syslibG6\"6#,&I\"xGF'\"\" \"*&^#F+F+I\"yGF'F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&polarGI(_sys libG6\"6$\"\"&-I'arctanG6$%*protectedGF$6##\"\"%\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"$\"\"\"*&\"\"%F$^#F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'arctanG6$%*protectedGI(_syslibG6\"6##\"\"%\"\"$" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "3.4.11. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z:=16*sqrt(3)-I*16; p olar(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#;\"\"\")\"\"$#F%\"\"# F%F%*&F$F%^#F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&polarGI(_sys libG6\"6$\"#K,$*&#\"\"\"\"\"'F+I#PiG%*protectedGF+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 62 "3.4.12. Gy\303\266kvon\303\241s komplex sz\303\241mb\303 \263l." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "3.4.13. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "z:='z'; i: ='i'; w:=16*sqrt(3)-I*16; solve(z^5=w,z); z1:=w^(1/5);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 30 "r:=abs(w); phi:=argument(w);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 71 "r^(1/5)*(cos(phi/5+i*2*Pi/5)+I*sin(phi/5+ i*2*Pi/5))$i=0..4; evalf(%);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 40 "solve(z^5=1); map(z->evalf(z*z1),[%]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"iG6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#;\"\"\")\"\"$#F%\"\"#F%F%*&F$F%^# F%F%!\"\"" }}{PARA 7 "" 1 "" {TEXT 208 37 "Warning, solutions may have been lost" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$),&*&\"#;\"\"\")\"\"$#F' \"\"#F'F'*&F&F'^#F'F'!\"\"#F'\"\"&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&#\"\"\"\"\"'F%I#PiG%*protect edGF%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'*&)\"#K#\"\"\"\"\"&F',&- I$sinG6$%*protectedGI(_syslibG6\"6#,$*&#\"\"(\"#:F'I#PiGF-F'F'F'*&,$^# F'F'F'-I$cosGF,F0F'!\"\"F'*&F$F',&-F+6#,$*&#\"\"#F5F'F6F'F'F'*&F9F'-F; F@F'F'F'*&F$F',&-F+6#,$*&#\"\"%F5F'F6F'F'F<*&F9F'-F;FJF'F'F'*&F$F',&*& #F'FDF')\"\"$FTF'F<*&FTF'F9F'F!\"*\" \"\"*&$\"+e#p04#!#5F'^#F'F'!\"\",&$\"+gGtM\")F+F'*&$\"+:44F=F&F'F,F'F' ,&$\"+^'*G'[\"F&F-*&$\"+77EQ8F&F'F,F'F',&$\"+330KF&F'F,F'F-" }}{PARA 7 "" 1 "" {TEXT 208 37 "Warning, solutions may have been lost" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\",(#F#\"\"%!\"\"*&F%F#)\"\"&#F#\"\"#F#F#*(*&F%F#^# F#F#F#)F,F+F#),&F*F#*$F)F#F#F+F#F#,(F%F'F(F'*(F.F#F0F#),&F*F#F3F'F+F#F #,(F%F'F(F'*(,$F.F#F#F0F#F6F#F',(F%F'F(F#*(F:F#F0F#F1F#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7',&$\"+\"zV!*)>!\"*\"\"\"*&$\"+l#p04#!#5F'^#F'F' !\"\",&$\"+eGtM\")F+F'*&$\"+:44F=F&F'F,F'F',&$\"+^'*G'[\"F&F-*&$\"+77E Q8F&F'F,F'F',&$\"+230KF&F'F,F'F-" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 37 "3.4.14. Az algebra alapt\30 3\251tele." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 69 "f:=(x-1)^2*(x-2); f:=expand(f); solve(f,x); solve(x ^3=1,x); r:=[%];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&),&I\"xG6\"\"\" \"F'!\"\"\"\"#F',&F%F'F)F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**$)I\" xG6\"\"\"$\"\"\"F(*&\"\"%F()F%\"\"#F(!\"\"*&\"\"&F(F%F(F(F,F-" }} {PARA 7 "" 1 "" {TEXT 208 37 "Warning, solutions may have been lost" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#\"\"\"F$" }}{PARA 7 "" 1 "" {TEXT 208 37 "Warning, solutions may have been lost" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%\"\"\",&#F#\"\"#!\"\"*&*&F%F#^#F#F#F#)\"\"$F%F#F#,&F% F'*&,$F)F#F#F+F#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"\",&#F#\"\"# !\"\"*&*&F%F#^#F#F#F#)\"\"$F%F#F#,&F%F'*&,$F)F#F#F+F#F'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->3.4.15. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->3.4.16. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "- >3.4.17. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "3.4.18. Fe ladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "3.4.19. Feladat." }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "3.4.20. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "3.4.21. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->3.4.22. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->3.4.23. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->3.4.24. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "- >3.4.25. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "3.4.26. Fe ladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "3.4.27. Feladat." }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "3.4.28. Kvaterni\303\263k." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "`&+`:=(p,q)->[p[1]+q[1],p[2]+q[2]];\n" }{MPLTEXT 1 0 77 "`&*`:=( p,q)->[p[1]*q[1]-conjugate(q[2])*p[2],q[2]*p[1]+p[2]*conjugate(q[1])]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"pG6\"I\"qGF%F%6$I)operatorGF% I&arrowGF%F%7$,&&F$6#\"\"\"F.&F&F-F.,&&F$6#\"\"#F.&F&F2F.F%F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"pG6\"I\"qGF%F%6$I)operatorGF%I&ar rowGF%F%7$,&*&&F$6#\"\"\"F/&F&F.F/F/*&-I*conjugateG%*protectedG6#&F&6# \"\"#F/&F$F7F/!\"\",&*&F-F/F6F/F/*&F9F/-F36#F0F/F/F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "p:=[a+I*b,c+I*d]; p&+[0,0]; p&+[-a- I*b,-c-I*d];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"aG6\"\"\"\"*&^# F&F&I\"bGF%F&F&,&I\"cGF%F&*&F(F&I\"dGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"aG6\"\"\"\"*&^#F&F&I\"bGF%F&F&,&I\"cGF%F&*&F(F&I\" dGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p&*[1,0]; [1,0]&*p;\n" }{MPLTEXT 1 0 2 "\n " }{MPLTEXT 1 0 60 "q:=[(a-I*b)/(a^2+b^2+c^2+d^2),(-c-I*d)/(a^2+b^2+c^ 2+d^2)];\n" }{MPLTEXT 1 0 55 "p&*q;evalc(%);simplify(%); q&*p;evalc(%) ;simplify(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"aG6\"\"\"\"*&^ #F&F&I\"bGF%F&F&,&I\"cGF%F&*&F(F&I\"dGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"aG6\"\"\"\"*&^#F&F&I\"bGF%F&F&,&I\"cGF%F&*&F(F&I\" dGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$*&,&I\"aG6\"\"\"\"*&,$^#F' F'F'I\"bGF&F'!\"\"F',**$)F%\"\"#F'F'*$)F+F0F'F'*$)I\"cGF&F0F'F'*$)I\"d GF&F0F'F'F,*&,&F5F,*&F)F'F8F'F,F'F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&*(,&I\"aG6\"\"\"\"*&^#F(F(I\"bGF'F(F(F(,&F&F(*&,$F*F(F(F+F(!\"\"F (,**$)F&\"\"#F(F(*$)F+F3F(F(*$)I\"cGF'F3F(F(*$)I\"dGF'F3F(F(F/F(*&-I*c onjugateG%*protectedG6#*&,&F8F/*&F.F(F;F(F/F(F0F/F(,&F8F(*&F*F(F;F(F(F (F/,&*(F%F(FBF(F0F/F(*&FDF(-F>6#*&F,F(F0F/F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,(*&,&*$)I\"aG6\"\"\"#\"\"\"F+*$)I\"bGF)F*F+F+F+,*F&F+F ,F+*$)I\"cGF)F*F+F+*$)I\"dGF)F*F+F+!\"\"F+*&F1F+F/F6F+*&F4F+F/F6F+,**& ,&*&F(F+F2F+F6*&F.F+F5F+F+F+F/F6F+*(F2F+F(F+F/F6F+*(F5F+F.F+F/F6F6*&^# F+F+,(*&,&*&F.F+F2F+F6*&F(F+F5F+F6F+F/F6F+*(F5F+F(F+F/F6F+*(F2F+F.F+F/ F6F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&*(,&I\"aG6\"\"\"\"*&^#F(F(I\"bGF'F(F(F(,&F&F(*& ,$F*F(F(F+F(!\"\"F(,**$)F&\"\"#F(F(*$)F+F3F(F(*$)I\"cGF'F3F(F(*$)I\"dG F'F3F(F(F/F(*(-I*conjugateG%*protectedG6#,&F8F(*&F*F(F;F(F(F(,&F8F/*&F .F(F;F(F/F(F0F/F/,&*(F,F(F0F/FAF(F(*(FCF(F0F/-F>6#F%F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "7$,&*&,&*$)I\"aG6\"\"\"#\"\"\"F+*$)I\"bGF)F*F+F+F+ ,*F&F+F,F+*$)I\"cGF)F*F+F+*$)I\"dGF)F*F+F+!\"\"F+*&,&F0F+F3F+F+F/F6F+ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "z:='z';w:='w';z1:='z1';p:=[z,w];p1:=[z1,w 1];p2:=[z2,w2];\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 46 "p&*(p1&*p2); expand(%);(p&*p1)&*p2;expand(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "I \"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"wG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I#z1G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$I\"zG6\"I\"wG F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$I#z1G6\"I#w1GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$I#z2G6\"I#w2GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$ ,&*&I\"zG6\"\"\"\",&*&I#z1GF&F'I#z2GF&F'F'*&-I*conjugateG%*protectedG6 #I#w2GF&F'I#w1GF&F'!\"\"F'F'*&-F.6#,&*&F*F'F1F'F'*&F2F'-F.6#F+F'F'F'I \"wGF&F'F3,&*&F%F'F7F'F'*&F " 0 "" {MPLTEXT 1 0 41 "p&*(p1&+p2);expand( %);(p&*p1)&+(p&*p2);\n" }{MPLTEXT 1 0 41 "(p1&+p2)&*p;expand(%);(p1&*p )&+(p2&*p);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&*&I\"zG6\"\"\"\",&I #z1GF&F'I#z2GF&F'F'F'*&-I*conjugateG%*protectedG6#,&I#w1GF&F'I#w2GF&F' F'I\"wGF&F'!\"\",&*&F%F'F0F'F'*&F3F'-F-6#F(F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,**&I\"zG6\"\"\"\"I#z1GF&F'F'*&F%F'I#z2GF&F'F'*&-I*conj ugateG%*protectedG6#I#w1GF&F'I\"wGF&F'!\"\"*&F1F'-F-6#I#w2GF&F'F2,**&F %F'F0F'F'*&F%F'F6F'F'*&F1F'-F-6#F(F'F'*&F1F'-F-6#F*F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,**&I\"zG6\"\"\"\"I#z1GF&F'F'*&F%F'I#z2GF&F'F'*&- I*conjugateG%*protectedG6#I#w1GF&F'I\"wGF&F'!\"\"*&F1F'-F-6#I#w2GF&F'F 2,**&F%F'F0F'F'*&F%F'F6F'F'*&F1F'-F-6#F(F'F'*&F1F'-F-6#F*F'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "7$,&*&I\"zG6\"\"\"\",&I#z1GF&F'I#z2GF&F' F'F'*&-I*conjugateG%*protectedG6#I\"wGF&F',&I#w1GF&F'I#w2GF&F'F'!\"\", &*&F(F'F0F'F'*&F1F'-F-6#F%F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,**& I\"zG6\"\"\"\"I#z1GF&F'F'*&F%F'I#z2GF&F'F'*&-I*conjugateG%*protectedG6 #I\"wGF&F'I#w1GF&F'!\"\"*&F,F'I#w2GF&F'F2,**&F0F'F(F'F'*&F0F'F*F'F'*&- F-6#F%F'F1F'F'*&F9F'F4F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,**&I\"z G6\"\"\"\"I#z1GF&F'F'*&F%F'I#z2GF&F'F'*&-I*conjugateG%*protectedG6#I\" wGF&F'I#w1GF&F'!\"\"*&F,F'I#w2GF&F'F2,**&F0F'F(F'F'*&F0F'F*F'F'*&-F-6# F%F'F1F'F'*&F9F'F4F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "j :=[0,1]; j&*j; [z,0]&+([w,0]&*j);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$ \"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$I\"zG6\"I\"wGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "k:=[0,I]; k&*k; i:=[I,0]; i&*i; [a,0]&+([b,0]&*i)&+([ c,0]&*j)&+([d,0]&*k);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!^#\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$^#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$!\"\" \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"aG6\"\"\"\"*&^#F&F&I\"bG F%F&F&,&I\"cGF%F&*&F(F&I\"dGF%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "p:=[a+I*b,c+I*d]; evalc([x,0]&*p); evalc(p&*[x,0]);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"aG6\"\"\"\"*&^#F&F&I\"bGF%F&F &,&I\"cGF%F&*&F(F&I\"dGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&*&I \"xG6\"\"\"\"I\"aGF&F'F'*(^#F'F'F%F'I\"bGF&F'F',&*&F%F'I\"cGF&F'F'*(F* F'F%F'I\"dGF&F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&*&I\"xG6\"\"\" \"I\"aGF&F'F'*(^#F'F'F%F'I\"bGF&F'F',&*&F%F'I\"cGF&F'F'*(F*F'F%F'I\"dG F&F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "j&*[z,0]; [z,0]&* j;" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!-I*conjugateG%*protectedG6# I\"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!I\"zG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "i&*j; j&*k; k&*i; j&*i; k&*j; i&*k; \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!^#\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "7$^#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\" !\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!,$^#\"\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "7$,$^#\"\"\"!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "i:='i'; j:='j'; k:='k';\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 76 "C2toR4:=q->evalc(Re(q[1])+Im(q[1])*i+Re(q[2])*j+Im(q[2])*k); q:=C2 toR4(p);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"iG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"jG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"kG6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"qG6\"F%6$I)operatorGF%I&arrowGF%F %-I&evalcG6$%*protectedGI(_syslibGF%6#,*-I#ReGF,6#&F$6#\"\"\"F5*&-I#Im GF,F2F5I\"iGF%F5F5*&-F16#&F$6#\"\"#F5I\"jGF%F5F5*&-F8F " 0 "" {MPLTEXT 1 0 104 "R4toC2:=q->[q-coeff(q,i)*i-coeff(q,j)*j-co eff(q,k)*k+I*coeff(q,i),coeff(q,j)+I*coeff(q,k)]; R4toC2(q);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"qG6\"F%6$I)operatorGF%I&arrowGF%F %7$,,F$\"\"\"*&-I&coeffG%*protectedG6$F$I\"iGF%F+F1F+!\"\"*&-F.6$F$I\" jGF%F+F6F+F2*&-F.6$F$I\"kGF%F+F:F+F2*&^#F+F+F-F+F+,&F4F+*&F " 0 "" {MPLTEXT 1 0 82 "qIm:=q->coeff(q,i)*i+coeff(q,j)*j+coeff(q,k)*k;qRe:=q ->q-qIm(q); qRe(q); qIm(q);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\" qG6\"F%6$I)operatorGF%I&arrowGF%F%,(*&-I&coeffG%*protectedG6$F$I\"iGF% \"\"\"F/F0F0*&-F,6$F$I\"jGF%F0F4F0F0*&-F,6$F$I\"kGF%F0F8F0F0F%F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"qG6\"F%6$I)operatorGF%I&arrowGF%F %,&F$\"\"\"-I$qImGF%F#!\"\"F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\" aG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I\"bG6\"\"\"\"I\"iGF%F&F&*& I\"cGF%F&I\"jGF%F&F&*&I\"dGF%F&I\"kGF%F&F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "qconjugate:=q->qRe(q)-qIm(q); qconjugate(q);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"qG6\"F%6$I)operatorGF%I&arrowGF%F %,&-I$qReGF%F#\"\"\"-I$qImGF%F#!\"\"F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*I\"aG6\"\"\"\"*&I\"bGF$F%I\"iGF$F%!\"\"*&I\"cGF$F%I\"jG F$F%F)*&I\"dGF$F%I\"kGF$F%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "q; qconjugate(q); qconjugate(%); q+qconjugate(q); q-qconjugate(q );" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*I\"aG6\"\"\"\"*&I\"bGF$F%I\"iGF$ F%F%*&I\"cGF$F%I\"jGF$F%F%*&I\"dGF$F%I\"kGF$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*I\"aG6\"\"\"\"*&I\"bGF$F%I\"iGF$F%!\"\"*&I\"cGF$F%I\"jG F$F%F)*&I\"dGF$F%I\"kGF$F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*I\"aG6 \"\"\"\"*&I\"bGF$F%I\"iGF$F%F%*&I\"cGF$F%I\"jGF$F%F%*&I\"dGF$F%I\"kGF$ F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"aG6\"F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",(*(\"\"#\"\"\"I\"bG6\"F%I\"iGF'F%F%*(F$ F%I\"cGF'F%I\"jGF'F%F%*(F$F%I\"dGF'F%I\"kGF'F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "q1:=a1+b1*i+c1*j+d1*k; q2:=a2+b2*i+c2*j+d2*k; \n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 75 "q1+q2; collect(%,[i,j,k]); \+ `&+`:=(q1,q2)->collect(q1+q2,[i,j,k]); q1&+q2;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 50 "`&*`:=proc(q1,q2) local a1,a2,b1,b2,c1,c2,d1,d2;\n" }{MPLTEXT 1 0 58 "a1:=qRe(q1);a2:=qRe(q2);b1:=coeff(q1,i);b2:=coeff(q 2,i);\n" }{MPLTEXT 1 0 66 "c1:=coeff(q1,j);c2:=coeff(q2,j);d1:=coeff(q 1,k);d2:=coeff(q2,k);\n" }{MPLTEXT 1 0 56 "(a1*a2-b1*b2-c1*c2-d1*d2)+( a1*b2+a2*b1+c1*d2-d1*c2)*i+\n" }{MPLTEXT 1 0 62 "(a1*c2+c1*a2+d1*b2-b1 *d2)*j+(a1*d2+d1*a2+b1*c2-c1*b2)*k;end;\n" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 9 "q1&*q2;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 53 "qcon jugate(q1&+q2); qconjugate(q1)&+qconjugate(q2);\n" }{MPLTEXT 1 0 2 "\n " }{MPLTEXT 1 0 66 "qconjugate(q1&*q2);qconjugate(q2)&*qconjugate(q1); expand(%%-%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*I#a1G6\"\"\"\"*&I# b1GF$F%I\"iGF$F%F%*&I#c1GF$F%I\"jGF$F%F%*&I#d1GF$F%I\"kGF$F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",*I#a2G6\"\"\"\"*&I#b2GF$F%I\"iGF$F%F%*& I#c2GF$F%I\"jGF$F%F%*&I#d2GF$F%I\"kGF$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2I#a1G6\"\"\"\"*&I#b1GF$F%I\"iGF$F%F%*&I#c1GF$F%I\"jGF$F %F%*&I#d1GF$F%I\"kGF$F%F%I#a2GF$F%*&I#b2GF$F%F(F%F%*&I#c2GF$F%F+F%F%*& I#d2GF$F%F.F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&,&I#b1G6\"\"\"\"I #b2GF&F'F'I\"iGF&F'F'*&,&I#c2GF&F'I#c1GF&F'F'I\"jGF&F'F'*&,&I#d2GF&F'I #d1GF&F'F'I\"kGF&F'F'I#a1GF&F'I#a2GF&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I#q1G6\"I#q2GF%F%6$I)operatorGF%I&arrowGF%F%-I(collectG6$%*pro tectedGI(_syslibGF%6$,&F$\"\"\"F&F17%I\"iGF%I\"jGF%I\"kGF%F%F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",,*&,&I#b1G6\"\"\"\"I#b2GF&F'F'I\"iGF&F' F'*&,&I#c2GF&F'I#c1GF&F'F'I\"jGF&F'F'*&,&I#d2GF&F'I#d1GF&F'F'I\"kGF&F' F'I#a1GF&F'I#a2GF&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I#q1G6\"I#q2 GF%6*I#a1GF%I#a2GF%I#b1GF%I#b2GF%I#c1GF%I#c2GF%I#d1GF%I#d2GF%F%F%C+>F( -I$qReGF%6#F$>F)-F36#F&>F*-I&coeffG%*protectedG6$F$I\"iGF%>F+-F:6$F&F= >F,-F:6$F$I\"jGF%>F--F:6$F&FD>F.-F:6$F$I\"kGF%>F/-F:6$F&FK,0*&F(\"\"\" F)FQFQ*&F*FQF+FQ!\"\"*&F,FQF-FQFS*&F.FQF/FQFS*&,**&F(FQF+FQFQ*&F)FQF*F QFQ*&F,FQF/FQFQ*&F.FQF-FQFSFQF=FQFQ*&,**&F(FQF-FQFQ*&F,FQF)FQFQ*&F.FQF +FQFQ*&F*FQF/FQFSFQFDFQFQ*&,**&F(FQF/FQFQ*&F.FQF)FQFQ*&F*FQF-FQFQ*&F,F QF+FQFSFQFKFQFQF%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*&I#a1G6\"\" \"\"I#a2GF%F&F&*&I#b1GF%F&I#b2GF%F&!\"\"*&I#c1GF%F&I#c2GF%F&F+*&I#d1GF %F&I#d2GF%F&F+*&,**&F$F&F*F&F&*&F'F&F)F&F&*&F-F&F1F&F&*&F0F&F.F&F+F&I \"iGF%F&F&*&,**&F$F&F.F&F&*&F-F&F'F&F&*&F0F&F*F&F&*&F)F&F1F&F+F&I\"jGF %F&F&*&,**&F$F&F1F&F&*&F0F&F'F&F&*&F)F&F.F&F&*&F-F&F*F&F+F&I\"kGF%F&F& " }}{PARA 11 "" 1 "" {XPPMATH 20 ",,I#a1G6\"\"\"\"I#a2GF$F%*&,&I#b1GF$ F%I#b2GF$F%F%I\"iGF$F%!\"\"*&,&I#c2GF$F%I#c1GF$F%F%I\"jGF$F%F,*&,&I#d2 GF$F%I#d1GF$F%F%I\"kGF$F%F," }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&,&I#b 1G6\"!\"\"I#b2GF&F'\"\"\"I\"iGF&F)F)*&,&I#c2GF&F'I#c1GF&F'F)I\"jGF&F)F )*&,&I#d2GF&F'I#d1GF&F'F)I\"kGF&F)F)I#a1GF&F)I#a2GF&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*&I#a1G6\"\"\"\"I#a2GF%F&F&*&I#b1GF%F&I#b2GF%F&!\" \"*&I#c1GF%F&I#c2GF%F&F+*&I#d1GF%F&I#d2GF%F&F+*&,**&F$F&F*F&F&*&F'F&F) F&F&*&F-F&F1F&F&*&F0F&F.F&F+F&I\"iGF%F&F+*&,**&F$F&F.F&F&*&F-F&F'F&F&* &F0F&F*F&F&*&F)F&F1F&F+F&I\"jGF%F&F+*&,**&F$F&F1F&F&*&F0F&F'F&F&*&F)F& F.F&F&*&F-F&F*F&F+F&I\"kGF%F&F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*&I #a1G6\"\"\"\"I#a2GF%F&F&*&I#b1GF%F&I#b2GF%F&!\"\"*&I#c1GF%F&I#c2GF%F&F +*&I#d1GF%F&I#d2GF%F&F+*&,**&F'F&F)F&F+*&F$F&F*F&F+*&F0F&F.F&F&*&F-F&F 1F&F+F&I\"iGF%F&F&*&,**&F-F&F'F&F+*&F$F&F.F&F+*&F)F&F1F&F&*&F0F&F*F&F+ F&I\"jGF%F&F&*&,**&F0F&F'F&F+*&F$F&F1F&F+*&F-F&F*F&F&*&F)F&F.F&F+F&I\" kGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 63 "3.4.29. Kvaterni\303\263k abszol\303\272t \303\251rt\303\251ke." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "qabs:=q->sqrt(qRe(q)^2+coeff(q,i)^2+coeff(q,j)^2+coeff(q,k)^2); \n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 28 "qabs(q); q&*qconjugate(q); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"qG6\"F%6$I)operatorGF%I&arr owGF%F%-I%sqrtG6$%*protectedGI(_syslibGF%6#,**$)-I$qReGF%F#\"\"#\"\"\" F5*$)-I&coeffGF,6$F$I\"iGF%F4F5F5*$)-F96$F$I\"jGF%F4F5F5*$)-F96$F$I\"k GF%F4F5F5F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$),**$)I\"aG6\"\"\"# \"\"\"F**$)I\"bGF(F)F*F**$)I\"cGF(F)F*F**$)I\"dGF(F)F*F*#F*F)F*" }} {PARA 11 "" 1 "" {XPPMATH 20 ",**$)I\"aG6\"\"\"#\"\"\"F(*$)I\"bGF&F'F( F(*$)I\"cGF&F'F(F(*$)I\"dGF&F'F(F(" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->3. 4.30. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "*3.4.31. Vekt ori" }{TEXT 207 8 "\303\241" }{TEXT 207 9 "lis szorz" }{TEXT 207 8 "\3 03\241" }{TEXT 207 2 "s." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 98 "*3. 4.32. Kvaterni\303\263k \303\251s a h\303\241romdimenzi\303\263s eukli d\303\251szi t\303\251r." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "q1:=b1*i+c1*j+d1*k; q2:=b2*i+c2*j+d 2*k; q3:=b3*i+c3*j+d3*k;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 55 "sca larprod:=(q1,q2)->-qRe(q1&*q2); scalarprod(q1,q2);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 54 "vectorprod:=(q1,q2)->qIm(q1&*q2); vectorprod(q1 ,q2);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 58 "mixedprod:=(q1,q2,q3)- >scalarprod(q1,vectorprod(q2,q3));\n" }{MPLTEXT 1 0 22 "mixedprod(q1,q 2,q3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I#b1G6\"\"\"\"I\"iGF%F&F &*&I#c1GF%F&I\"jGF%F&F&*&I#d1GF%F&I\"kGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I#b2G6\"\"\"\"I\"iGF%F&F&*&I#c2GF%F&I\"jGF%F&F&*&I#d2 GF%F&I\"kGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I#b3G6\"\"\"\"I \"iGF%F&F&*&I#c3GF%F&I\"jGF%F&F&*&I#d3GF%F&I\"kGF%F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I#q1G6\"I#q2GF%F%6$I)operatorGF%I&arrowGF%F%,$-I $qReGF%6#-I#&*GF%F#!\"\"F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I# b1G6\"\"\"\"I#b2GF%F&F&*&I#c1GF%F&I#c2GF%F&F&*&I#d1GF%F&I#d2GF%F&F&" } }{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I#q1G6\"I#q2GF%F%6$I)operatorGF%I&a rrowGF%F%-I$qImGF%6#-I#&*GF%F#F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,(*&,&*&I#c1G6\"\"\"\"I#d2GF'F(F(*&I#d1GF'F(I#c2GF'F(!\"\"F(I\"iGF'F(F (*&,&*&F+F(I#b2GF'F(F(*&I#b1GF'F(F)F(F-F(I\"jGF'F(F(*&,&*&F4F(F,F(F(*& F&F(F2F(F-F(I\"kGF'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I#q1G6\"I #q2GF%I#q3GF%F%6$I)operatorGF%I&arrowGF%F%-I+scalarprodGF%6$F$-I+vecto rprodGF%6$F&F'F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I#b1G6\"\"\" \",&*&I#c2GF%F&I#d3GF%F&F&*&I#d2GF%F&I#c3GF%F&!\"\"F&F&*&I#c1GF%F&,&*& F,F&I#b3GF%F&F&*&I#b2GF%F&F*F&F.F&F&*&I#d1GF%F&,&*&F5F&F-F&F&*&F)F&F3F &F.F&F&" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "*3.4.33. A szorz" }{TEXT 207 8 "\303\24 1" }{TEXT 207 21 "sok geometriai jelent" }{TEXT 207 8 "\303\251" } {TEXT 207 3 "se." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 15 "*3.4.34. Fo rgat" }{TEXT 207 11 "\303\241sok" }{TEXT 207 1 "." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 15 "*3.4.35. A skal" }{TEXT 207 8 "\303\241" }{TEXT 207 14 "ris es vektori" }{TEXT 207 8 "\303\241" }{TEXT 207 9 "lis szor z" }{TEXT 207 8 "\303\241" }{TEXT 207 21 "s geometriai alkalmaz" } {TEXT 207 8 "\303\241" }{TEXT 207 4 "sai." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 51 "* 3.4.36. Okt\303\241vok vagy Cayley-sz\303\241mok." }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->3.4.37. Feladat." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->3.4.38. Feladat." }}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 17 "*3.4.39. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "3.4.40. Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 20 "bbi \+ feladatok megold" }{TEXT 207 8 "\303\241" }{TEXT 207 7 "sokkal." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "3.4.41. Tov" }{TEXT 207 8 "\303 \241" }{TEXT 207 14 "bbi feladatok." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%# %?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 24 "4. V\303\251ges halmazo k" }}{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\241fe lm\303\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 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 }