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

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

Answer
Let x ∈ B
There can be two cases, x ∈ A or x ∉ A.
Case -1 x ∈ A
As x ∈ A and x ∈ B so that x ∈ ( A ∩ B)
Given that A∩ B = A ∩ C, so that
x ∈ ( A ∩ C)
x ∈ A and x ∈ C
x ∈ B then x ∈ C so that B ⊂ C.
Case -2, x ∉ A
We have already assumed that x ∈ B.
Hence, x ∈ (A ∪ B)
Given that A ∪ B = A ∪ C, so that
⇒ x ∈ (A ∪ C)
⇒x ∈ A or x ∈ C
But we assumed that x ∉ A Hence x ∈ C
x ∈ B then x ∈ C so that B ⊂ C.
So in both cases B ⊂ C. ...(1)
Similarly, we can prove C ⊂ B ...(2)
From equation (1) and (2)
B= C