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in Sets, Relations and Functions by (50.8k points)

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)

1 Answer

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Best answer

Given as

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

(i) A × (B ∪ C) = (A × B) ∪ (A × C)

Let us consider the LHS  (B ∪ C)

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

= {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}

Then, RHS

(A × B) = {1, 2, 3} × {4}

= {(1, 4), (2, 4), (3, 4)}

(A × C) = {1, 2, 3} × {5}

= {(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) A × (B ∩ C) = (A × B) ∩ (A × C)

Let us consider the LHS: (B ∩ C)

(B ∩ C) = ∅ (Here, no common element)

A × (B ∩ C) = {1, 2, 3} × ∅

= ∅

Then, RHS

(A × B) = {1, 2, 3} × {4}

= {(1, 4), (2, 4), (3, 4)}

(A × C) = {1, 2, 3} × {5}

= {(1, 5), (2, 5), (3, 5)}

(A × B) ∩ (A × C) = ∅

∴ LHS = RHS

(iii) A × (B − C) = (A × B) − (A × C)

Let us consider the LHS: (B − C)
(B − C) = ∅

A × (B − C) = {1, 2, 3} × ∅

= ∅

Then, RHS

(A × B) = {1, 2, 3} × {4}

= {(1, 4), (2, 4), (3, 4)}

(A × C) = {1, 2, 3} × {5}

= {(1, 5), (2, 5), (3, 5)}

(A × B) − (A × C) = ∅

Thus, LHS = RHS

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