Question

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

Answer 1

It is given that f : A × B → B × A is defined as f
(a, b) = (b, a)
Now let us consider $(a_1,b_1),(a_2,b_2)$ ∈ A × B
Such that $f(a_1,b_1)=f(a_2,b_2)$
⇒ (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)
⇒ f is onto.
$\textbf{Therefore, f is bijective.}$