Intersection[{a,b,c,d},{c,a,e,f}]

Union[{a,b,c,d},{c,a,e,f}]

(*Системи уравнения*)

Solve[{a x+y==0,2x+(a-1)y==1},{x,y}]

Solve[{x^2+y^2==1,x+3 y==0},{x,y}]

x+y /. %

Solve[{x+2y-z==7,-x+3y+4z==5,2x+6y+9z==11}]

Eliminate[{a x+y==0,2x+(a-1)y==1},y]

Eliminate[{a x+y==0,2x+(a-1)y==1},x]

Solve[{x==1,x==a},x]

Reduce[{x==1,x==a},x]

a==1&&x==1

Reduce[{a x+b==0},x]

(b==0&&a==0)||(a≠0&&x==-(b/a))

Reduce[{a x^2+b==0},x]

(b==0&&a==0)||(a≠0&&(x==-((I (Sqrt[b]))/(Sqrt[a]))||x==((I (Sqrt[b]))/(Sqrt[a]))))

Solve[{a x+b y==1,x-y==2},{x,y}]

{{x->-((-1-2 b)/(a+b)),y->-2-((-1-2 b)/(a+b))}}

Solve[{x^2+y^2==1,x+ y==a},{x,y}]

{{x->(1/2) (a-(Sqrt[2-(a^2)])),y->(1/2) (a+(Sqrt[2-(a^2)]))},{x->(1/2) (a+(Sqrt[2-(a^2)])),y->(1/2) (a-(Sqrt[2-(a^2)]))}}

x^3+y^3 /.%/.a->7/10

{{x->(1/2) (a-(Sqrt[2-(a^2)])),y->(1/2) (a+(Sqrt[2-(a^2)]))},{x->(1/2) (a+(Sqrt[2-(a^2)])),y->(1/2) (a-(Sqrt[2-(a^2)]))}}

%%%/.{a->1,b->-2}

Solve[x^3-7x+3==0]

N[%]

{{x->2.3976615408922592` +0.` I},{x->0.44080771150488274` +4.440892098500626`*^-16 I},{x->-2.838469252397142`+0.` I}}

Plot[x^3-7x+3,{x,-3,2.5}]

NSolve[x^3-7x+3==0]

{{x->-2.8384692523971418`},{x->0.44080771150488296`},{x->2.397661540892259`}}

Solve[{x^3+y^3==x y,x+ y+x y==1},{x,y}]

N[%]

{{x->0.32638236713558594`,y->0.5078608171783358`},{x->0.5078608171783358`,y->0.326382367135586`},{x->-1.745431807680516`-1.20180340326451` I,y->-1.7454318076805162`+1.201803403264509` I},{x->-1.745431807680516`+1.20180340326451` I,y->-1.7454318076805162`-1.201803403264509` I},{x->-0.1716897844764448`-1.1462557248970762` I,y->-0.17168978447644512`+1.1462557248970762` I},{x->-0.1716897844764448`+1.1462557248970762` I,y->-0.17168978447644512`-1.1462557248970762` I}}

Table[2 a[i]+a[i-1]==a[i+1],{i,3}]

{a[0]+2 a[1]==a[2],a[1]+2 a[2]==a[3],a[2]+2 a[3]==a[4]}

Solve[%,{a[1],a[2],a[3]}]

{{a[1]->(1/12) (-5 a[0]+a[4]),a[2]->((a[0])/6)+((a[4])/6),a[3]->-((a[0])/12)+((5 a[4])/12)}}

Table[2(Subscript[a, i])+(Subscript[a, i-1])==(Subscript[a, i+1]),{i,3}]

{(Subscript[a, 0])+2 (Subscript[a, 1])==(Subscript[a, 2]),(Subscript[a, 1])+2 (Subscript[a, 2])==(Subscript[a, 3]),(Subscript[a, 2])+2 (Subscript[a, 3])==(Subscript[a, 4])}

Solve[%,{(Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3])}]

{{(Subscript[a, 1])->(1/12) (-5 (Subscript[a, 0])+(Subscript[a, 4])),(Subscript[a, 2])->((Subscript[a, 0])/6)+((Subscript[a, 4])/6),(Subscript[a, 3])->-((Subscript[a, 0])/12)+((5 (Subscript[a, 4]))/12)}}

A={{3,1},{2,-5}}

MatrixForm[%]

(({{3, 1}, {2, -5}}))

A={{3,1},{2,-5}};
A.{x,y}=={7,8}

{3 x+y,2 x-5 y}=={7,8}

Solve[%,{x,y}]

{{x->(43/17),y->-(10/17)}}

LogicalExpand[%%]

2 x-5 y==8&&3 x+y==7

Solve[ArcSin[x]==a,x]

{{x->Sin[a]}}

{{x->Sin[a]}}

Solve[Sin[x]==a,x]

Reduce[Sin[x]==a,x]

C[1]∈Integers&&(x==π-ArcSin[a]+2 π C[1]||x==ArcSin[a]+2 π C[1])

Reduce[x^2-7y^2==1&&x>0&&y>0,{x,y},Integers]

C[1]∈Integers&&C[1]≥1&&x==(1/2) ((((8-3 (Sqrt[7])))^(C[1]))+(((8+3 (Sqrt[7])))^(C[1])))&&y==-(((((8-3 (Sqrt[7])))^(C[1]))-(((8+3 (Sqrt[7])))^(C[1])))/(2 (Sqrt[7])))