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Question

Let A and B be sets. Show that f:A×BB×A such that f(a,b)=(b,a) is bijective function.

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Solution

f:A×BB×A is defined as f(a,b)=(b,a).

Let (a1,b1),(a2,b2)A×B such that f(a1,b1)=f(a2,b2).
(b1,a1)=(b2,a2)
b1=b2) and (a1=a2)
(a1,b1)=(a2,b2)
f is one-one.
Now, let (b,a)B×A be any element.
Then, there exists (a,b)A×B such that

f(a,b)=(b,a). [By definition of f]

f is onto.
Hence, f is bijective.

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