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Question

Let A and B be two sets. Show that the sets A×B and B×A have an element in common if the sets A and B have an element in common.

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Solution

Let (a, b) be an arbitary element of (A×B)(B×A). Then,

(a,b)ϵ(B×A). Then,

(a,b)ϵ(A×B)(B×A)

(a,b)ϵA×B and (a,b)ϵB×A

(a,ϵ A and b ϵ B)and (a ϵ B and b ϵ A)

(a,ϵ A and a ϵ B)and (b ϵ A and b ϵ B)

a ϵ A B and b ϵ A B

Hence, the sets A×B and B×A have an element in common iff the sets A and have an element in common.


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