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Question
Let A and B be sets. Show that f: A × B → B × A such that (a, b) = (b, a) is bijective function.
Solution
f: A × B → B × A is defined as f(a, b) = (b, a).
Let `(a_1, b_1), (a_2, b_2) in A xx B " such that " f(a_1, b_1) = (a_2, b_2)`
`=> (b_1 , a_1) = (b_2, a_2)`
`=> b_1 = b_2 and a_1 = a_2`
`=> (a_1, b_1) =(a_2, b_2)`
∴ f is injective.
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 bijective.
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