Let x ∈ A ∩ (B - C) ⇒ x ∈ A and x ∈(B - C)
⇒ x ∈ A and (x ∈ B and x ∉ C)
⇒ (x ∈ A and x ∈ B) and (x ∈ A and x ∉ C)
= x ∈ (A ∩ B and x ∈ (A ∩ C)
⇒ x ∈(A ∩ B) - (A ∩ C)
A ∩ (B - C) ⊆ (A ∩ B) - (A ∩ C) .............. (1)
Thus, Let
x ∈ {(A ∩ B) - (A ∩ C)}
⇒ x ∈ (A ∩ B) and x ∈ (A ∩ C)
⇒ x ∈ (A ∩ B) and x ∈ C
⇒ x ∈ A and (x ∈ B and x ∈ C)
⇒ x ∈ A and x ∈ (B - C)
⇒ x ∈ A ∩ (B - C)
(A ∩ B - (A ∩ C) ⊆ A ∩ (B - C) ................ (2)
Now, from equation (1) and (2)
A ∩ (B - C) = (A ∩ B) - (A ∩ C)
Hence Proved.
