Kalkulus I.J\303\241rai AntalEzek a programok csak szeml\303\251ltet\303\251sre szolg\303\241lnak1. Halmazok2. Sz\303\241mokrestart;2.1. Val\303\263s sz\303\241mok2.1.1. Test.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.2. P\303\251lda.&+(0,0):=0; &+(0,1):=1; &+(1,0):=1; &+(1,1):=0;
&*(0,0):=0; &*(0,1):=0; &*(1,0):=0; &*(1,1):=1;
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.3. P\303\251ld\303\241k.`&+`:=(x,y)->irem(x+y,5); `&*`:=(x,y)->irem(x*y,5); 3&+4; 3&*4;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn* 2.1.4. Algebrai strukt\303\272r\303\241k.isgrupoid:=proc(G::set,f::procedure) local x,y;
for x in G do for y in G do if not f(x,y) in G then return false fi;
od; od; true; end;
neutral:=proc(G::set,f::procedure) local x,y,s,S;
if not isgrupoid(G,f) then return NULL fi;
for x in G do s:=true; for y in G do
if f(x,y)<>y or f(y,x)<>y then s:=false; break; fi;
od; if s then return x fi; od; NULL end;
G:={a,b,c};neutral(G,(x,y)->y);neutral(G,(x,y)->y);
neutral({0,1,2},(x,y)->irem(x+y,3));
issemigroup:=proc(G::set,f::procedure) local x,y,z;
if not isgrupoid(G,f) then return false fi;
for x in G do for y in G do for z in G do
if f(x,f(y,z))<>f(f(x,y),z) then return false fi;
od; od; od; true end;
issemigroup({a,b,c},(x,y)->x);
issemigroup({true,false},(x,y)-> x implies y);
isgroup:=proc(G::set,f::procedure) local x,y,n,i;
if not isgrupoid(G,f) then return false fi;
if not issemigroup(G,f) then return false fi;
n:=neutral(G,f); if n=NULL then return false fi;
for x in G do i:=false; for y in G do
if f(x,y)=n and f(y,x)=n then i:=true; break fi;
od; if i=false then return false fi; od; true; end;
isgroup({0,1,2},(x,y)->irem(x+y,3));
iscommutative:=proc(G::set,f::procedure) local x,y;
if not isgrupoid(G,f) then return false fi;
for x in G do for y in G do
if f(x,y)<>f(y,x) then return false fi;
od; od; true; end;
iscommutative({0,1,2},(x,y)->irem(x+y,3));
isabeliangroup:=proc(G::set,f::procedure)
isgroup(G,f) and iscommutative(G,f) end;
iscommutative({0,1,2},(x,y)->irem(x+y,3));
isleftdistributive:=proc(R::set,f::procedure,g::procedure) local x,y,z;
if not isgrupoid(R,f) then return false fi;
if not isgrupoid(R,g) then return false fi;
for x in R do for y in R do for z in R do
if g(x,f(y,z))<>f(g(x,y),g(x,z)) then return false fi;
od; od; od; true end;isrightdistributive:=proc(R::set,f::procedure,g::procedure) local x,y,z;
if not isgrupoid(R,f) then return false fi;
if not isgrupoid(R,g) then return false fi;
for x in R do for y in R do for z in R do
if g(f(y,z),x)<>f(g(y,x),g(z,x)) then return false fi;
od; od; od; true end;isring:=proc(R::set,f::procedure,g::procedure)
isabeliangroup(R,f) and issemigroup(R,g)
and isleftdistributive(R,f,g) and isrightdistributive(R,f,g) end;iscommutativering:=proc(R::set,f::procedure,g::procedure)
isring(R,f,g) and iscommutative(R,g) end;isringwithunity:=proc(R::set,f::procedure,g::procedure)
isring(R,f,g) and neutral(R,g)<>NULL end;isskewfield:=proc(R::set,f::procedure,g::procedure) local n;
n:=neutral(R,f); if n=NULL then return false fi;
isring(R,f,g) and isgroup(R minus {n},g) end;isfield:=proc(R::set,f::procedure,g::procedure) local n;
n:=neutral(R,f); if n=NULL then return false fi;
isring(R,f,g) and isabeliangroup(R minus {n},g) end;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn* 2.1.5. P\303\251ld\303\241k.iff:=(x,y)->evalb((x implies y) and (y implies x));
X:={true,false}; isabeliangroup(X,(x,y)->iff(x,y));
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn* 2.1.6. P\303\251ld\303\241k.X:={a,b,c};with(combinat,powerset);P:=powerset(X);isgroup(P,(x,y)->symmdiff(x,y));
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn* 2.1.7. P\303\251ld\303\241k.isring({0},(x,y)->0,(x,y)->0);isring(P,(x,y)->symmdiff(x,y),(x,y)->{});LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn* 2.1.8. P\303\251ld\303\241k.iscommutativering(P,(x,y)->symmdiff(x,y),(x,y)->x intersect y);
isringwithunity(P,(x,y)->symmdiff(x,y),(x,y)->x intersect y);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn* 2.1.9. P\303\251ld\303\241k.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.10. Rendezett test.abs(7.4); abs(-3); abs(0); signum(7.4); signum(-3); signum(0);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.11. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.12. P\303\251ld\303\241k.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.13. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.14. Val\303\263s sz\303\241mok.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.15. Term\303\251szetes sz\303\241mok.inc:=x->x+1;0;inc(%);inc(%);inc(%);inc(%);
dec:=x->x-1;4;dec(%);dec(%);dec(%);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.16. Rekurzi\303\263t\303\251tel.n:=16;twopower:=1;for i to n do twopower:=twopower*2; od;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.17. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.18. Sorozatok.i:='i';j:='j';$3..9;i^2$i=2/3..10/3;x[i]$i=3..8;{j^i$j=i..8}$i=2..4;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.19. P\303\251lda.cat("ab","bcc");cat("bcc","ab");evalb(%=%%);"ab"||"bcc";with(StringTools,Generate):Generate(4,"abc");LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.20. Motiv\303\241ci\303\263: tov\303\241bbi rekurz\303\255v defin\303\255ci\303\263k.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.21. \303\201ltal\303\241nos rekurzi\303\263 t\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.22. Fibonacci-sz\303\241mok.Fib:=proc(n::nonnegint) option remember; if n<2 then n else Fib(dec(n))+Fib(dec(dec(n))) fi end;interface(verboseproc=3):
print(Fib);
Fib(3);print(Fib);
Fib(7);print(Fib);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.23. Szorzatok \303\251s \303\266sszegek.prodfrom1:=proc(x,n) if n<1 then 1 elif n=1 then x[1] else prodfrom1(x,dec(n))*x[n] fi end;
prodfrom1(y,5);
product(y[i],i=0..7);
sum(x[j],j=4.5..8.7);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.24. Az \303\241ltal\303\241nos disztributivit\303\241s t\303\251tele.A:=sum(a[i],i=1..4);B:=sum(b[j],j=1..5);A*B;expand(%);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.25. Faktori\303\241lis, binomi\303\241lis egy\303\274tthat\303\263.0!; 1!; 2!; 3!; 4!; 5!; 6!;binomial(6,0);binomial(6,1);binomial(6,2);binomial(6,3);binomial(6,4);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.26. Binomi\303\241lis t\303\251tel.(x+y)^6;expand(%);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn* 2.1.27. K\303\266vetkezm\303\251ny.sum(binomial(n,k),k=0..n);sum(binomial(n,k)*(-1)^k,k=0..n);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.28. Eg\303\251sz sz\303\241mok.type(5,integer); type(-3,integer); type(0,integer); type(3.14,integer);
type(5,posint); type(-3,negint); type(0,posint); type(0,nonnegint); type(0,nonposint);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.29. Hatv\303\241nyoz\303\241s eg\303\251sz kitev\305\221vel.x^(-5); x^(n+m); expand(%); (x^4)^(-5); (x*y)^5;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.30. Racion\303\241lis sz\303\241mok.type(5/7,rational); type(0,rational);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.31. Archim\303\251d\303\251szi tulajdons\303\241g.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.32. \303\201ll\303\255t\303\241s.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.33. Eg\303\251sz r\303\251sz, marad\303\251k.floor(3.14); ceil(3.14); ceil(-3.14); Rmod:=proc(x::realcons,y::realcons) if y=0 then x else x-floor(x/y)*y fi; end;
Rmod(5,0); Rmod(3.1415,2.78);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.34. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.35. T\303\251tel: gy\303\266kvon\303\241s.root[2](2); evalf(%); sqrt(2); root[12](2); evalf(%);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.36. K\303\266vetkezm\303\251ny.* 2.1.37. \303\201ll\303\255t\303\241s.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn* 2.1.38. \303\201ll\303\255t\303\241s.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.1.39. B\305\221v\303\255tett val\303\263s sz\303\241mok.infinity; -infinity; evalb(5<infinity); evalb(5<-infinity);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2. Megsz\303\241ml\303\241lhat\303\263 halmazok.restart;2.2.1. Halmazok ekvivalenci\303\241ja.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.2. \303\201ll\303\255t\303\241s.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.3. Megjegyz\303\251s.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.4. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.5. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.6. V\303\251ges \303\251s v\303\251gtelen halmazok.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.7. Karakterisztikus f\303\274ggv\303\251nyek.X:={a,b,c,d,e}; Y:={a,c,e}; chi:=x-> if x in Y then 1 elif x in X then 0 else FAIL fi;chi(a);chi(b);chi(1);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.8. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.9 Skatulya elv.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.10. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.11. Megsz\303\241ml\303\241lhat\303\263 halmazok.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.12. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.13. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.14. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.15. T\303\251tel.for k from 0 do for m from 0 to k do n:=k-m: T:=time(); while time()<T+1 do od; print([m,n],(k*(k+1)/2)+m); od; od;2.2.16. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.17. K\303\266vetkezm\303\251ny.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.18. T\303\251tel.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.19. K\303\266vetkezm\303\251ny.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.2.20. Cantor t\303\251tele.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3. Komplex sz\303\241mokrestart;2.3.1. Komplex sz\303\241mok.`&+`:=proc(z,w) [z[1]+w[1],z[2]+w[2]] end;
`&*`:=proc(z,w) [z[1]*w[1]-z[2]*w[2],z[1]*w[2]+z[2]*w[1]] end;
[x,y]&+[0,0]; [x,y]&+[-x,-y]; [x,y]&*[1,0];
[x,y]&*[x/(x^2+y^2),-y/(x^2+y^2)]; simplify(%);
[0,1]&*[0,1];
Complex(3,5); z:=3+5*I; w:=-2-6*I; z*w; Re(z); Im(z); conjugate(z);
z:='z';w:='w'; conjugate(z);
conjugate(conjugate(z));conjugate(z+w);conjugate(1/z);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.2. P\303\251lda.64/(3^(1/2)+I); evalc(%);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.3. Komplex sz\303\241mok abszol\303\272t \303\251rt\303\251ke.z:=x+I*y;abs(z);evalc(%);evalc(1/(x+I*y));evalc(conjugate(z)/abs(z)^2);signum(3+4*I); signum(-5); signum(0);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.4. Komplex sz\303\241mok argumentuma \303\251s trigonometrikus alakja.polar(x+I*y); op(1,%); op(2,%%); polar(3+4*I); evalc(%); argument(3+I*4);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.5. P\303\251lda.z:=16*sqrt(3)-I*16; polar(z);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.6. Gy\303\266kvon\303\241s komplex sz\303\241mb\303\263l.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.7. P\303\251lda.z:='z'; i:='i'; w:=16*sqrt(3)-I*16; solve(z^5=w,z); z1:=w^(1/5);
r:=abs(w); phi:=argument(w);
r^(1/5)*(cos(phi/5+i*2*Pi/5)+I*sin(phi/5+i*2*Pi/5))$i=0..4; evalf(%);
solve(z^5=1); map(z->evalf(z*z1),[%]);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.8. B\305\221v\303\255tett komplex sz\303\241mok.z:=infinity+I*infinity; w:=infinity-I*infinity; evalb(z=w);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.9. Kvaterni\303\263k.`&+`:=(p,q)->[p[1]+q[1],p[2]+q[2]];
`&*`:=(p,q)->[p[1]*q[1]-conjugate(q[2])*p[2],q[2]*p[1]+p[2]*conjugate(q[1])];p:=[a+I*b,c+I*d]; p&+[0,0]; p&+[-a-I*b,-c-I*d];
p&*[1,0]; [1,0]&*p;
q:=[(a-I*b)/(a^2+b^2+c^2+d^2),(-c-I*d)/(a^2+b^2+c^2+d^2)];
p&*q;evalc(%);simplify(%); q&*p;evalc(%);simplify(%);
z:='z';w:='w';z1:='z1';p:=[z,w];p1:=[z1,w1];p2:=[z2,w2];
p&*(p1&*p2);expand(%);(p&*p1)&*p2;expand(%);
p&*(p1&+p2);expand(%);(p&*p1)&+(p&*p2);
(p1&+p2)&*p;expand(%);(p1&*p)&+(p2&*p);
j:=[0,1]; j&*j; [z,0]&+([w,0]&*j);k:=[0,I]; k&*k; i:=[I,0]; i&*i; [a,0]&+([b,0]&*i)&+([c,0]&*j)&+([d,0]&*k);
p:=[a+I*b,c+I*d]; evalc([x,0]&*p); evalc(p&*[x,0]);
j&*[z,0]; [z,0]&*j;i&*j; j&*k; k&*i; j&*i; k&*j; i&*k;
i:='i'; j:='j'; k:='k';
C2toR4:=q->evalc(Re(q[1])+Im(q[1])*i+Re(q[2])*j+Im(q[2])*k); q:=C2toR4(p);
R4toC2:=q->[q-coeff(q,i)*i-coeff(q,j)*j-coeff(q,k)*k+I*coeff(q,i),coeff(q,j)+I*coeff(q,k)]; R4toC2(q);
qIm:=q->coeff(q,i)*i+coeff(q,j)*j+coeff(q,k)*k;qRe:=q->q-qIm(q); qRe(q); qIm(q);
qconjugate:=q->qRe(q)-qIm(q); qconjugate(q);
q; qconjugate(q); qconjugate(%); q+qconjugate(q); q-qconjugate(q);q1:=a1+b1*i+c1*j+d1*k; q2:=a2+b2*i+c2*j+d2*k;
q1+q2; collect(%,[i,j,k]); `&+`:=(q1,q2)->collect(q1+q2,[i,j,k]); q1&+q2;
`&*`:=proc(q1,q2) local a1,a2,b1,b2,c1,c2,d1,d2;
a1:=qRe(q1);a2:=qRe(q2);b1:=coeff(q1,i);b2:=coeff(q2,i);
c1:=coeff(q1,j);c2:=coeff(q2,j);d1:=coeff(q1,k);d2:=coeff(q2,k);
(a1*a2-b1*b2-c1*c2-d1*d2)+(a1*b2+a2*b1+c1*d2-d1*c2)*i+
(a1*c2+c1*a2+d1*b2-b1*d2)*j+(a1*d2+d1*a2+b1*c2-c1*b2)*k;end;
q1&*q2;
qconjugate(q1&+q2); qconjugate(q1)&+qconjugate(q2);
qconjugate(q1&*q2);qconjugate(q2)&*qconjugate(q1); expand(%%-%);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.10. Kvaterni\303\263k \303\251s a h\303\241rom dimenzi\303\263s euklid\303\251szi t\303\251r.q1:=b1*i+c1*j+d1*k; q2:=b2*i+c2*j+d2*k; q3:=b3*i+c3*j+d3*k;
scalarprod:=(q1,q2)->-qRe(q1&*q2); scalarprod(q1,q2);
vectorprod:=(q1,q2)->qIm(q1&*q2); vectorprod(q1,q2);
mixedprod:=(q1,q2,q3)->scalarprod(q1,vectorprod(q2,q3));
mixedprod(q1,q2,q3);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.11. Kvaterni\303\263k abszol\303\272t \303\251rt\303\251ke.qabs:=q->sqrt(qRe(q)^2+coeff(q,i)^2+coeff(q,j)^2+coeff(q,k)^2);
qabs(q); q&*qconjugate(q);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.3.12. A skal\303\241r- vektor- \303\251s vegyes szorzat geometriai jelent\303\251se.qabs(q1&+q2)^2-qabs(q1-q2)^2;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4. Polinomokrestart;2.4.1. Jel\303\266l\303\251s.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.2. Polinomok \303\251s racion\303\241lis t\303\266rtf\303\274ggv\303\251nyek.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.3. A marad\303\251kos oszt\303\241s t\303\251tele polinomokra.f:=3*x^3+6*x^2+7*x-8; g:=x^2+8*x-2; q:=quo(f,g,x); r:=rem(f,g,x);
g*q+r; expand(%);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.4. K\303\266vetkezm\303\251ny: Horner-elrendez\303\251s.rem(f,x-5,x); subs(x=5,f);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.5. K\303\266vetkezm\303\251ny: gy\303\266kt\303\251nyez\305\221 lev\303\241laszt\303\241sa.f:=(x-1)^2*(x-2); f:=expand(f); quo(f,x-2,x);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.6. K\303\266vetkezm\303\251ny.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.7. K\303\266vetkezm\303\251ny.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.8. Polinomok egy\303\251rtelm\305\261 fel\303\255r\303\241sa.coeff(f,x,0); coeff(f,x,1); coeff(f,x,2); coeff(f,x,3); coeff(f,x,4); lcoeff(f,x); lcoeff(0,x);degree(f); degree(0);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.9. A marad\303\251kos oszt\303\241s egy\303\251rtelm\305\261s\303\251ge.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.10. T\303\266bbsz\303\266r\303\266s gy\303\266k\303\266k.subs(x=1,f); quo(f,x-1,x); subs(x=1,f); quo(f,(x-1)^2,x);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.11. Az algebra alapt\303\251tele.solve(f,x); solve(x^3=1,x); r:=[%];
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.12. Gy\303\266kt\303\251nyez\305\221s el\305\221\303\241ll\303\255t\303\241s.map(y->x-y,r); convert(%,`*`); evalc(%);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn* 2.4.13. Lagrange-interpol\303\241ci\303\263.CurveFitting[PolynomialInterpolation]([0,1,2,3],[0,3,1,3],x);
CurveFitting[PolynomialInterpolation]([0,1,2,3],[0,3,1,3],x,form=Lagrange);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2.4.14. T\303\266bbv\303\241ltoz\303\263s polinomok \303\251s racion\303\241lis t\303\266rtf\303\274ggv\303\251nyek.f:=5*x^2*y; type(f,monomial); type(f,polynom); degree(f);
f:=f+x*y^3; type(f,monomial); type(f,polynom); degree(f);
f/(2*x*y-5*x^2*y^2); type(f,ratpoly);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn3. Hat\303\241r\303\251rt\303\251kLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn4. Differenci\303\241lsz\303\241m\303\255t\303\241sLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn5. Integr\303\241lsz\303\241m\303\255t\303\241sLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn