Question

if A union B = A intersection B, prove that A = B

Answer 1

Suppose x∈Ax∈A, then x∈A∪Bx∈A∪B. So then x∈A∩Bx∈A∩B.
Thus x∈Bx∈B and thus A⊆BA⊆B

Suppose x∈Bx∈B, then x∈A∪Bx∈A∪B. So then x∈A∩Bx∈A∩B.
Thus x∈Ax∈A and thus B⊆AB⊆A


Therefore, A=B



《 Or, In another way 》


A⊂A∪B=A∩B⊂A

and

B⊂A∪B=A∩B⊂B.


Therefore,

A=A∪B=A∩B=B.


__________________

Hope it helps :)

Answer 2

We have,

A−(A∩B)A−(A∩B)

=A∩(A∩B)c=A∩(A∩B)c

[∵[∵ For two non-empty sets XX and Y,Y,

X−Y=X∩Yc,X−Y=X∩Yc, ‘c’ denotes complement set.]]

=A∩(Ac∪Bc)=A∩(Ac∪Bc) [De-Morgan's Law.]

=(A∩Ac)∪(A∩Bc)=(A∩Ac)∪(A∩Bc) [Distributive Law.]

=∅∪(A−B)=∅∪(A−B)

=A−B.=A−B. [Identity Law.]†

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