UPSC NDA Math Model Paper 10

Let x ∈ C
Suppose x ∈ A
Here, x is common elements of A and C.
Therefore, x ∈ A ∩ C
Given: A ∩ B = A ∩ C
⇒ x ∈ A ∩ B (∵ x ∈ A ∩ C)
Thus, x ∈ B
∴ B = C
Again,
Let x ∈ C
Suppose x ∉ A
Therefore, x ∈ A ∪ C
Given: A ∪ B = A ∪ C
⇒ x ∈ A ∪ B (∵ x ∈ A ∪ C)
Thus, x ∈ B
Hence C ⊆ B .... (1)
Similarly, we can show that C ⊆ A .... (2)
From (1) and (2) we get B = C