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

It is given that f : A × B → B × A is defined as f (a, b) = (b, a)

Now let us consider (a₁, b₁), (a₂, b₂) ϵ 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) ϵ B × A be any element.

Then, there exists (a, b) ϵ A × B such that f(a, b) = (b, a)

⇒ f is onto.

Therefore, f is bijective.