Prove that:
(A ∩ B) x C = (A x C) ∩ (B x C)
To prove: (A ∩ B) × C = (A × C) ∩ (B×C)
Proof:
Let (x, y) be an arbitrary element of (A ∩ B) × C.
(x, y) ∈ (A ∩ B) × C
Since, (x, y) are elements of Cartesian product of (A ∩ B)× C
x ∈ (A ∩ B) and y ∈ C
(x ∈ A and x ∈B) and y ∈ C
(x ∈ A and y ∈ C) and (x ∈ Band y ∈ C)
(x, y) ∈ A × C and (x, y) ∈ B × C
(x, y) ∈ (A × C) ∩ (B × C) …1
Let (x, y) be an arbitrary element of (A × C) ∩ (B × C).
(x, y) ∈ (A × C) ∩ (B × C)
(x, y) ∈ (A × C) and (x, y) ∈ (B × C)
(x ∈A and y ∈ C) and (x ϵ Band y ∈ C)
(x ∈A and x ∈ B) and y ∈ C
x ∈ (A ∩ B) and y ∈ C
(x, y) ∈ (A ∩ B) × C …2
From 1 and 2, we get: (A ∩ B) × C = (A × C) ∩ (B × C)