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

According to the question,

A ∪ B = A ∪ C

And,

A ∩ B = A ∩ C

To show,

B = C

Let us assume,

x ∈ B

So,

x ∈ A ∪ B

x ∈ A ∪ C

Hence,

x ∈ A or x ∈ C

When x ∈ A, then,

x ∈ B

x ∈ A ∩ B

As, A ∩ B = A ∩ C

So, x ∈ A ∩ C

x ∈ A or x ∈ C

x ∈ C

B ⊂ C

Similarly, it can be shown that C ⊂ B

Hence, B = C