Question

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

Answer 1

Solution :

$f:A\times B\to B\times A$ is defined as $f(a,b)=(b,a)$

Let $(a_1,b_1),(a_2,b_2)$ & $A\times B$ such that

$f(a_1,b_1)=f(a_2,b_2)$

$\Rightarrow (b_1,a_1)=(b_2,a_2)$

$\Rightarrow b_1=b_2$ & $a_1=a_2$

$\Rightarrow (a_1,b_1)=(a_2,b_2)$

$\therefore$ f is one - one

Now, Let $(b,a)ϵB\times A$ be any element

Then, there exists $(a,b)ϵA\times B$ such that

$f(a,b)=(b,a)...$ [By difinition of f]

f is outo. Hence f is bijective