Question

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

Answer 1

f: \(A \times B → B \times A\) is defined as f(a, b) = (b, a).

Let \((a_1,b_1)\), \((a_2,b_2)\) \(∈ A\times B\) such that \(f(a_1,b_1)=f(a_2,b_2)\).
\(⇒ (b_1,a_1)=(b_2,a_2)\)
\(⇒ b_1=b_2 \text{ and } a_1=a_2\)
\(⇒ (a_1,b_1)=(a_2,b_2)\)
∴ 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.