Question

Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C.

Answer 1

The sets are given as,

A∪B=A∪C

A∩B=A∩C

Let,

x∈B

Here, one of the elements of set B is x.

So,

x∈A∪B x∈A∪C

x∈A

And,

x∈C

Take,

x∈A

And,

x∈B

x∈A∩B x∈A∩C

So,

x∈A

And,

x∈C

Hence, an element belongs to Bthen, it will belong to C, that is,

B⊂C(1)

Now let,

x∈C

Hence,

x∈A∪C x∈A∪B

x∈A

And,

x∈B

Taking,

x∈A

And,

x∈C

x∈A∩C x∈A∩B

So,

x∈A

And,

x∈C

Hence, an element belongs to C then, it will belong to B, that is,

C⊂B(2)

From equations (1) and (2),

B = C

Therefore,

B=C