Given: A = {a, b, c, d,}, B = {c, d, e} and C = {d, e, f, g}

(i) Need to prove: A × (B ∩ C) = (A × B) ∩ (A × C)

Left hand side,

(B ∩ C) = {d, e}

⇒ A × (B ∩ C) = {(a, d), (a, e), (b, d), (b, e), (c, d), (c, e), (d, d), (d, e)}

Right hand side,

(A × B) = {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e), (c, c), (c, d), (c, e), (d, c), (d, d), (d, e)}

(A × C) = {(a, d), (a, e), (a, f), (a, g), (b, d), (b, e), (b, f), (b, g), (c, d), (c, e), (c, f), (c, g), (d, d), (d, e), (d, f), (d, g)}

Now,

(A × B) ∩ (A × C) = {(a, d), (a, e), (b, d), (b, e), (c, d), (c, e), (d, d), (d, e)}

Here, right hand side and left hand side are equal.

That means, A × (B ∩ C) = (A × B) ∩ (A × C) [Proved]

(ii) Need to prove: A × (B – C) = (A × B) – (A × C)

Left hand side,

(B – C) = {c}

⇒ A × (B – C) = {(a, c), (b, c), (c, c), (d, c)}

Right hand side,

(A × B) = {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e), (c, c), (c, d), (c, e), (d, c), (d, d), (d, e)}

(A × C) = {(a, d), (a, e), (a, f), (a, g), (b, d), (b, e), (b, f), (b, g), (c, d), (c, e), (c, f), (c, g), (d, d), (d, e), (d, f), (d, g)}

Therefore, (A × B) – (A × C) = {(a, c), (b, c), (c, c), (d, c)}

Here, right hand side and left hand side are equal.

That means, A × (B – C) = (A × B) – (A × C) [Proved]

(iii) Need to prove: (A × B) ∩ (B × A) = (A ∩ B) × (A ∩ B)

Left hand side,

(A × B) = {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e), (c, c), (c, d), (c, e), (d, c), (d, d), (d, e)}

(B × A) = {(c, a), (c, b), (c, c), (c, d), (d, a), (d, b), (d, c), (d, d), (e, a), (e, b), (e, c), (e, d)}

Now, (A × B) ∩ (B × A) = {(c, c), (c, d), (d, c), (d, d)}

Right hand side,

(A ∩ B) = {c, d}

So, (A ∩ B) × (A ∩ B) = {(c, c), (c, d), (d, c), (d, d)}

Here, right hand side and left hand side are equal.

That means, (A × B) ∩ (B × A) = (A ∩ B) × (A ∩ B) [Proved]