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{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 201 69 "Ezek a programok csak \+
szeml\303\251ltet\303\251sre szolg\303\241lnak." }}}{SECT 1 {PARA 3 ""
 0 "" {TEXT 205 11 "1. Halmazok" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0
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zetes sz\303\241mok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0
 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
206 32 "2.1. Peano-axi\303\263m\303\241k" }}{PARA 0 "" 0 "" {TEXT 201 
0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 34 "2.1.1.Peano-axi\303\263m
\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 77 "inc:=x->x+1;0;inc(%);inc(%);inc(%);inc(%);dec:=x->x
-1;4;dec(%);dec(%);dec(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6
\"F%6$I)operatorGF%I&arrowGF%F%,&F$\"\"\"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 "\"\"$" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"f*6#I\"xG6\"F%6$I)operatorGF%I&arrowGF%F%,&F$\"\"\"F*!\"\"F%F%F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "
\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G"
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 }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 
"" 0 "" {TEXT 207 26 "*2.1.3. Megjegyz\303\251s." }}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 35 
"2.1.4. Rekurzi\303\263t\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 ""
 }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "n:=16;twopower:=1;for i t
o n do twopower:=twopower*2; od;" }}{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 "\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#k" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
\"$c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$7&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"%C5" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%[?" }}{PARA 11 
"" 1 "" {XPPMATH 20 "\"%'4%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%#>)" }
}{PARA 11 "" 1 "" {XPPMATH 20 "\"&%Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"\"&oF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&Ob'" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 78 
"2.1.5. A term\303\251szetes sz\303\241mok egy\303\251rtelm\305\261s\3
03\251ge." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 
1 {PARA 5 "" 0 "" {TEXT 207 65 "2.1.6. A term\303\251szetes sz\303\241
mok l\303\251tez\303\251se." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" 
"%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 49 "2.1.7. Karakteriszti
kus f\303\274ggv\303\251nyek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "X:=\{a,b,c,d,e\}; Y:=\{a,c,e
\};\n" }{MPLTEXT 1 0 60 "chi:=x-> if x in Y then 1 elif x in X then 0 \+
else FAIL fi;\n" }{MPLTEXT 1 0 23 "chi(a); chi(b); chi(1);" }}{PARA 11
 "" 1 "" {XPPMATH 20 "<'I\"aG6\"I\"bGF$I\"cGF$I\"eGF$I\"dGF$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "<%I\"aG6\"I\"cGF$I\"eGF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF%I&arrowGF%F%@'-I#inG%*protec
tedG6$F$I\"YGF%\"\"\"-F+6$F$I\"XGF%\"\"!I%FAILGF,F%F%F%" }}{PARA 11 ""
 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I%FAILG%*protectedG" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 
17 "->2.1.8. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "2.1.9.
 Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->2.1.10. Feladat.
" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->2.1.11. Feladat." }}}
{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->2.1.12. Feladat." }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT 207 18 "->2.1.13. Feladat." }}}{SECT 1 {PARA 5 "
" 0 "" {TEXT 207 16 "2.1.14. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" 
{TEXT 207 18 "->2.1.15. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 
207 16 "2.1.16. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "2.1
.17. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "2.1.18. Felada
t." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->2.1.19. Feladat." }}}
{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "2.1.20. Feladat." }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT 207 16 "2.1.21. Feladat." }}}{SECT 1 {PARA 5 "" 
0 "" {TEXT 207 16 "2.1.22. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 
207 16 "2.1.23. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "2.1
.24. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "2.1.25. Felada
t." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "2.1.26. Feladat." }}}
{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "2.1.27. Feladat." }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT 207 16 "2.1.28. Feladat." }}}{SECT 1 {PARA 5 "" 
0 "" {TEXT 207 17 "*2.1.29. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT
 207 16 "2.1.30. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "2.
1.31. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*2.1.32. Fela
dat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*2.1.33. Feladat." }}}
{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*2.1.34. Feladat." }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT 207 18 "->2.1.35. Feladat." }}}{SECT 1 {PARA 5 "
" 0 "" {TEXT 207 18 "->2.1.36. Feladat." }}}{SECT 1 {PARA 5 "" 0 "" 
{TEXT 207 11 "2.1.37. 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 57 "2.2. M\305\261veletek term\303\2
51szetes sz\303\241mokkal" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 65 "2.2
.1. Term\303\251szetes sz\303\241mok \303\266sszead\303\241sa." }}
{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "sm:=n->if n=0 then m else inc(sm(dec(n))) fi;\n" }{MPLTEXT 1 0 
36 "m:=3;sm(6);sm(10);m:=5;sm(6);sm(10);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "f*6#I\"nG6\"F%6$I)operatorGF%I&arrowGF%F%@%/F$\"\"!I\"mGF%-I$incGF
%6#-I#smGF%6#-I$decGF%F#F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20
 "\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#:" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" 
{TEXT 207 20 "2.2.2. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 56 "2.2.3. Term\3
03\251szetes sz\303\241mok szorz\303\241sa." }}{PARA 0 "" 0 "" {TEXT 
201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pm:=n->if n=0 the
n 0 else pm(dec(n))+m fi;\n" }{MPLTEXT 1 0 36 "m:=3;pm(6);pm(10);m:=5;
pm(6);pm(10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%6$I)opera
torGF%I&arrowGF%F%@%/F$\"\"!F+,&-I#pmGF%6#-I$decGF%F#\"\"\"I\"mGF%F2F%
F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#I" }}{PARA 11 ""
 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#I" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"#]" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 20 "2.2.
4. T\303\251tel." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->2.2.5. Fe
ladat." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->2.2.6. Feladat." }}
}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 8 "2.2.7. F" }{TEXT 207 8 "\303\251
" }{TEXT 207 31 "lcsoport, csoport, kommutativit" }{TEXT 207 8 "\303\2
41" }{TEXT 207 2 "s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "isgrupoid:=proc(G::set,f::procedure) loca
l 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 18 "od; od; true; end;" }}
{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
$-F*F,F%OI&falseGF(F1F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
57 "rightneutrals:=proc(G::set,f::procedure) local x,y,s,S;\n" }
{MPLTEXT 1 0 54 "if not isgrupoid(G,f) then return false fi; S:=\{\};
\n" }{MPLTEXT 1 0 38 "for x in G do s:=true; for y in G do\n" }
{MPLTEXT 1 0 87 "  if f(y,x)<>y then s:=false; break fi; od; if s then
 S:=S union \{x\} fi; od; S end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 
0 39 "G:=\{a,b,c\};rightneutrals(G,(x,y)->x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6$'I\"GG6\"I$setG%*protectedG'I\"fGF&I*procedureGF(6&I
\"xGF&I\"yGF&I\"sGF&I\"SGF&F&F&C&@$4-I*isgrupoidGF&6$F%F*OI&falseGF(>F
0<\"?&F-F%I%trueGF(C%>F/F<?&F.F%F<@$0-F*6$F.F-F.C$>F/F8[@$F/>F0-I&unio
nGF(6$F0<#F-F0F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"aG6\"I\"bGF
$I\"cGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"aG6\"I\"bGF$I\"cGF$" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "leftneutrals:=proc(G::set,f
::procedure) local x,y,s,S;\n" }{MPLTEXT 1 0 54 "if not isgrupoid(G,f)
 then return false fi; S:=\{\};\n" }{MPLTEXT 1 0 38 "for x in G do s:=
true; for y in G do\n" }{MPLTEXT 1 0 87 "  if f(x,y)<>y then s:=false;
 break fi; od; if s then S:=S union \{x\} fi; od; S end;\n" }{MPLTEXT 
1 0 2 "\n" }{MPLTEXT 1 0 64 "G:=\{a,b,c\};leftneutrals(G,(x,y)->y);rig
htneutrals(G,(x,y)->y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"GG6\"
I$setG%*protectedG'I\"fGF&I*procedureGF(6&I\"xGF&I\"yGF&I\"sGF&I\"SGF&
F&F&C&@$4-I*isgrupoidGF&6$F%F*OI&falseGF(>F0<\"?&F-F%I%trueGF(C%>F/F<?
&F.F%F<@$0-F*6$F-F.F.C$>F/F8[@$F/>F0-I&unionGF(6$F0<#F-F0F&F&F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"aG6\"I\"bGF$I\"cGF$" }}{PARA 11 "" 
1 "" {XPPMATH 20 "<%I\"aG6\"I\"bGF$I\"cGF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "neutr
al:=proc(G::set,f::procedure) local x,y,s,S;\n" }{MPLTEXT 1 0 44 "if n
ot isgrupoid(G,f) then return NULL fi;\n" }{MPLTEXT 1 0 38 "for x in G
 do s:=true; for y in G do\n" }{MPLTEXT 1 0 54 "  if f(x,y)<>y or f(y,
x)<>y then s:=false; break fi;\n" }{MPLTEXT 1 0 42 "od; if s then retu
rn x fi; od; NULL end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 55 "G:=\{
a,b,c\};neutral(G,(x,y)->y);neutral(G,(x,y)->y);\n" }{MPLTEXT 1 0 2 "
\n" }{MPLTEXT 1 0 38 "neutral(\{0,1,2\},(x,y)->irem(x+y,3));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "f*6$'I\"GG6\"I$setG%*protectedG'I\"fGF&I*proce
dureGF(6&I\"xGF&I\"yGF&I\"sGF&I\"SGF&F&F&C%@$4-I*isgrupoidGF&6$F%F*OI%
NULLGF(?&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
&falseGF([@$F/OF-F8F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"aG6\"I
\"bGF$I\"cGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 53 "issemigroup:=proc(G::set,f::procedure) lo
cal x,y,z;\n" }{MPLTEXT 1 0 45 "if not isgrupoid(G,f) then return fals
e fi;\n" }{MPLTEXT 1 0 43 "for x in G do for y in G do for z in G do\n
" }{MPLTEXT 1 0 53 "  if f(x,f(y,z))<>f(f(x,y),z) then return false fi
;\n" }{MPLTEXT 1 0 23 "od; od; od; true end;\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 34 "issemigroup(\{a,b,c\},(x,y)->x);\n" }{MPLTEXT 1 0 2 "
\n" }{MPLTEXT 1 0 50 "issemigroup(\{true,false\},(x,y)-> x implies y);
\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'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(@$4-I,issemigroupGF&F6F7>F/-I(neutralGF
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eG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "iscommuta
tive:=proc(G::set,f::procedure) local x,y;\n" }{MPLTEXT 1 0 45 "if not
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do for y in G do\n" }{MPLTEXT 1 0 43 "  if f(x,y)<>f(y,x) then return \+
false fi;\n" }{MPLTEXT 1 0 20 "od; od; true; end;\n" }{MPLTEXT 1 0 2 "
\n" }{MPLTEXT 1 0 46 "iscommutative(\{0,1,2\},(x,y)->irem(x+y,3));\n" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"GG6\"I$setG%*protectedG'I\"fGF
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"" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "isabeliangroup:=proc(G::set,f::procedure)\n" }
{MPLTEXT 1 0 42 "isgroup(G,f) and iscommutative(G,f) end;\n" }{MPLTEXT
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{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "X:=\{a,b,c\}; with(combinat,powerse
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{PARA 11 "" 1 "" {XPPMATH 20 "7#I)powersetG6\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<*<\"<%I\"aG6\"I\"bGF&I\"cGF&<#F%<#F'<$F%F'<#F(<$F'F(<$F%
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 0 "" {TEXT 207 29 "2.2.12. P\303\251ld\303\241k." }}}{SECT 0 {PARA 5 
"" 0 "" {TEXT 207 29 "2.2.13. P\303\251ld\303\241k." }}{PARA 0 "" 0 ""
 {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "X:=\{true
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tectedGI%trueGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F
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{XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2
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{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->2.2.16. Feladat." }}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" 
{TEXT 207 18 "->2.2.17. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 
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{PARA 5 "" 0 "" {TEXT 207 18 "->2.2.19. Feladat." }}{EXCHG {PARA 0 "" 
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"->2.2.20. Feladat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" 
}}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->2.2.21. Feladat." }}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 
0 "" {TEXT 207 18 "->2.2.22. Feladat." }}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->2.
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{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "2.2.24. Feladat." }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT 207 16 "2.2.25. Feladat." }}}{SECT 1 {PARA 5 "" 
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{XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 56 "2.3.
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0 "" {MPLTEXT 1 0 15 "2<=5; evalb(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
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{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 
0 "" {TEXT 207 20 "2.3.2. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" 
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}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 20 "2.3.4. T\303\251tel." }}{EXCHG 
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{TEXT 207 17 "2.3.5. Sorozatok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "i:='i'; j:='j'; $3..9; i^2$i
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{XPPMATH 20 "I\"iG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"jG6\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6)\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*" }
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#;\"#D\"#O\"#\\\"#k<(\"#FF*\"$D\"\"$;#\"$V$\"$7&<'\"$c#\"$D'\"%'H\"\"%
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1 {PARA 5 "" 0 "" {TEXT 207 17 "->2.3.6. Feladat." }}}{SECT 0 {PARA 5 
"" 0 "" {TEXT 207 25 "->2.3.7. Feladat: tranzit" }{TEXT 207 8 "\303\25
5" }{TEXT 207 5 "v lez" }{TEXT 207 8 "\303\241" }{TEXT 207 3 "rt." }}
{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 
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{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "2.3.10. P\303\251lda." }}{PARA 0
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t(\"ab\",\"bcc\"); cat(\"bcc\",\"ab\"); evalb(%=%%); \"ab\"||\"bcc\";"
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{XPPMATH 20 "Q&bccab6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*pro
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 "> " 0 "" {MPLTEXT 1 0 33 "StringTools[Generate](4,\"abc\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "7]pQ%aaaa6\"Q%aaabF$Q%aaacF$Q%aabaF$Q%aa
bbF$Q%aabcF$Q%aacaF$Q%aacbF$Q%aaccF$Q%abaaF$Q%ababF$Q%abacF$Q%abbaF$Q%
abbbF$Q%abbcF$Q%abcaF$Q%abcbF$Q%abccF$Q%acaaF$Q%acabF$Q%acacF$Q%acbaF$
Q%acbbF$Q%acbcF$Q%accaF$Q%accbF$Q%acccF$Q%baaaF$Q%baabF$Q%baacF$Q%baba
F$Q%babbF$Q%babcF$Q%bacaF$Q%bacbF$Q%baccF$Q%bbaaF$Q%bbabF$Q%bbacF$Q%bb
baF$Q%bbbbF$Q%bbbcF$Q%bbcaF$Q%bbcbF$Q%bbccF$Q%bcaaF$Q%bcabF$Q%bcacF$Q%
bcbaF$Q%bcbbF$Q%bcbcF$Q%bccaF$Q%bccbF$Q%bcccF$Q%caaaF$Q%caabF$Q%caacF$
Q%cabaF$Q%cabbF$Q%cabcF$Q%cacaF$Q%cacbF$Q%caccF$Q%cbaaF$Q%cbabF$Q%cbac
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}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 ""
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}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Fib:=proc(n::nonnegint) opt
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{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 18 "Fib(7);print(Fib);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "f*6#'I\"nG6\"I*nonnegintG%*protectedGF&6#I)remembe
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"f*6#'I\"nG6\"I*nonnegintG%*protectedGF&6#I)rememberGF&E\\s%\"\"!F,\"
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{XPPMATH 20 "f*6#'I\"nG6\"I*nonnegintG%*protectedGF&6#I)rememberGF&E
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0 17 "prodfrom1(y,5);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 23 "produc
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F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "*,&I\"yG6\"6#\"\"\"F'&F$6#\"\"#F
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"*2&I\"yG6\"6#\"\"!\"\"\"&F$6#F(F(&F$6#\"\"#F(&F$6#\"\"$F(&F$6#\"\"%F(
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{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 
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0 "" {TEXT 207 16 "2.3.19. Feladat." }}{EXCHG {PARA 0 "" 0 "" 
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{PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" 
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{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "2.3.32. Feladat." }}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" 
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{PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" 
{TEXT 207 27 "2.3.35. Feladat: Catalan-sz" }{TEXT 207 8 "\303\241" }
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{PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" 
{TEXT 207 12 "2.3.38. Norm" }{TEXT 207 8 "\303\241" }{TEXT 207 8 "lala
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{PARA 5 "" 0 "" {TEXT 207 55 "2.3.39. A marad\303\251kos oszt\303\241s
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{SECT 0 {PARA 5 "" 0 "" {TEXT 207 49 "2.3.40. H\303\241nyados \303\251
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{XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 44 "2.3.
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0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "convert(17,base,3); c
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{PARA 11 "" 1 "" {XPPMATH 20 "\"&,+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
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{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 
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{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "2.3.50. Feladat." }}}{SECT 0 
{PARA 5 "" 0 "" {TEXT 207 16 "2.3.51. Feladat." }}{EXCHG {PARA 0 "" 0 
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