Show that A × (B ∩ C) = (A × B) ∩ (A × C). Where A= {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}

To show that A × (B ∩ C) = (A × B) ∩ (A × C), we need to prove that the two sets are equal.

First, let's find A × (B ∩ C):

A × (B ∩ C) = {(1, x) | x ∈ (B ∩ C)}

Since B ∩ C = {}, the intersection of B and C is empty. Therefore, A × (B ∩ C) is also empty.

Next, let's find (A × B) ∩ (A × C):

(A × B) ∩ (A × C) = {(1, x) | x ∈ B} ∩ {(1, y) | y ∈ C}

Substituting the values of A, B, and C, we get:

(A × B) ∩ (A × C) = {(1, x) | x ∈ {1, 2, 3, 4}} ∩ {(1, y) | y ∈ {5, 6}}

Expanding the sets, we have:

(A × B) ∩ (A × C) = {(1, 1), (1, 2), (1, 3), (1, 4)} ∩ {(1, 5), (1, 6)}

Taking the intersection of these two sets, we find:

(A × B) ∩ (A × C) = {(1, 1), (1, 2), (1, 3), (1, 4)} ∩ {(1, 5), (1, 6)} = {}

Since both A × (B ∩ C) and (A × B) ∩ (A × C) are empty sets, we can conclude that A × (B ∩ C) = (A × B) ∩ (A × C).

Therefore, we have shown that A × (B ∩ C) = (A × B) ∩ (A × C).