If A = {1, 2, 3}, B = {4}, c = {5}, then verify that:

i. A x ( B ∪ C) = (A x B) ∪ (A x C)


ii. A x (B ∩ C) = (A x B) ∩ (A x C)


iii. A x (B – C) = (A x B) – (A x C).

given A = {1, 2, 3}, B = {4} and C = {5}

(i) To prove: A × (B ∪ C) = (A × B) ∪ (A × C)


LHS: (B ∪ C) = {4, 5}


therefore A × (B ∪ C) = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}


RHS:


(A × B) = {(1, 4), (2, 4), (3, 4)}


(A × C) = {(1, 5), (2, 5), (3, 5)}


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


∴ LHS = RHS


(ii) To prove: A × (B ∩ C) = (A × B) ∩ (A × C)


LHS: (B ∩ C) = ∅ (No common element)


A × (B ∩ C) = ∅


RHS: (A × B) = {(1, 4), (2, 4), (3, 4)}


(A × C) = {(1, 5), (2, 5), (3, 5)}


(A × B) ∩ (A × C) = ∅


∴ LHS = RHS


(iii) To prove: A × (B − C) = (A × B) − (A × C)


LHS: (B − C) = ∅


A × (B − C) = ∅


RHS: (A × B) = {(1, 4), (2, 4), (3, 4)}


(A × C) = {(1, 5), (2, 5), (3, 5)}


(A × B) − (A × C) = ∅


∴ LHS = RHS


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