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