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For any two sets A and B, show that A × B and B × A have an element in common if and only if A and B have an element in common.
For any two sets A and B, show that A × B and B × A have an element in common if and only if A and B have an element in common.
CBSE | Class 11 | Excercise 2B | Maths | Relations | RS Aggarwal
For any two sets A and B, show that A × B and B × A have an element in common if and only if A and B have an element in common.

Answer : We know,

(A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A)

Here A and B have an element in common i.e., n(A ∩ B) = 1 = (B ∩ A)

So, n((A × B) ∩ (B × A)) = n((A ∩ B) × (B ∩ A)) = n(A ∩ B) × n(B ∩ A) = 1 × 1 = 1

That means, A × B and B × A have an element in common if and only if A and B have an element in common. [Proved]

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