Let A and B be sets. Show that f:

Question:

Let A and B be sets. Show that fA × B → × A such that (ab) = (ba) is bijective function.

Solution:

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

Let $\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right) \in \mathrm{A} \times \mathrm{B}$ such that $f\left(a_{1}, b_{1}\right)=f\left(a_{2}, b_{2}\right)$.

$\Rightarrow\left(b_{1}, a_{1}\right)=\left(b_{2}, a_{2}\right)$

$\Rightarrow b_{1}=b_{2}$ and $a_{1}=a_{2}$

$\Rightarrow\left(a_{1}, b_{1}\right)=\left(a_{2}, b_{2}\right)$

∴ f is one-one.

Now, let $(b, a) \in B \times A$ be any element.

Then, there exists $(a, b) \in A \times B$ such that $f(a, b)=(b, a)$. [By definition of $f$ ]

∴ f is onto.

Hence, f is bijective.

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