Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that:

(i) A × C ⊂ B × D
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)

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Solution

Given:
A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}

(i) A × C ⊂ B × D
LHS: A × C = {(1, 5), (1, 6), (2, 5), (2, 6)}
RHS: B × D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}
∴ A × C ⊂ B × D

(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
We have:
(B ∩ C) = $ϕ$
LHS: A × (B ∩ C) = $ϕ$
Now,
(A × B) = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
(A × C) = {(1, 5), (1, 6), (2, 5), (2, 6)}
RHS: (A × B) ∩ (A × C) = $ϕ$
∴ LHS = RHS

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Similar questions

A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}

D = {5, 6, 7, 8} .

(i) A \times (B \cap C) = (A \times B) \cap (A \times C)

(i i) A \times C

B \times D

A = {1, 2}

B = {1, 2, 3, 4}

C = {5, 6}

D = {5, 6, 7, 8}

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Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

(i) A × (B ∩ C) = (A × B) ∩ (A × C)

(ii) A × C is a subset of B × D

A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}

D = {5, 6, 7, 8}

A \times (B \cap C) = (A \times B) \cap (A \times C)

A \times C

B \times D

A = {1, 2}, B = {1, 2, 3, 4}

C = {5, 6}

A \times (B \cap C) = (A \times B) \cap (A \times C)

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