If $A = {1, 2, 3}$ , $B = {1, 2}$ and $C = {2, 3}$ which one of the following is correct?

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

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

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

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

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Solution

The correct option is C $(A \times B) \cup (B \times A) = (A \times B) \cup (B \times C)$
$(A \times B) =$ ${(1, 1), (1, 2,), (2, 1), (2, 2), (3, 1), (3, 2)}$
$(B \times A) =$ ${(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}$
$(B \times C) =$ ${(1, 2,), (1, 3), (2, 2), (2, 3)}$
$(C \times B) =$ ${(2, 1), (2, 2), (3, 1) (3, 2)}$
$(A \times C) =$ ${(2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}$
$(C \times A) =$ ${(2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}$
$∴ (A \times B) \cup (B \times A) = {(1, 1), (1, 2), (2, 2) (3, 1) (3, 2), (1, 3), (2, 3)}$
$∴ (A \times B) \cup (B \times A) = (A \times B) \cup (B \times C)$

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A \times (B \cap C) =

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

(i) $A \times (B \cup C) = (A \times B) \cup (A \times C)$

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(iii) $A \times (B - C) = (A \times B) - (A \times C)$

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