UPSC NDA Math Model Paper 10

Given, A and B are two disjoint sets.
⇒ (A ∩ B) = ϕ
Consider the statement "A - B = A - (A ∩ B)"
A - B = {x : x ∈ A, x ∉ B}
⇒ A - B = A ....(1)
Now, A - (A ∩ B) = A - ϕ
⇒ A - (A ∩ B) = A ....(2)
From (1) and (2), we have
A - B = A - (A ∩ B)
The statement " A - B = A - (A ∩ B)" is true.
Consider the statement "A - A' = B ∩ B"
A - A' = {x : x ∈ A, x ∉ B}
⇒ A - A' = A ....(3)
Now, B ∩ B = B ....(4)
From (3) and (4), we have
A - A' ≠ B ∩ B
The statement " A - A' = B ∩ B" is not true.
Consider the statement "A ∩ B = (A - B) ∩ B"
Let, A - B = {x : x ∈ A, x ∉ B}
⇒ A - B = A
⇒ (A - B) ∩ B = A∩ B
The statement "A ∩ B = (A - B) ∩ B" is true.
Hence, the statement A ∩ B = (A - B) ∩ B is true.