Consider \[{{\mathbf{f}}_{\mathbf{2}}}~=\text{ }\left\{ \left( \mathbf{2},~\mathbf{a} \right),\text{ }\left( \mathbf{3},~\mathbf{b} \right),\text{ }\left( \mathbf{4},~\mathbf{c} \right) \right\};~\mathbf{A}~=\text{ }\left\{ \mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4} \right\},~\mathbf{B}~=\text{ }\left\{ \mathbf{a},~\mathbf{b},~\mathbf{c} \right\}\]

Injectivity:

\[{{f}_{2}}~\left( 2 \right)~=\text{ }a\]

\[{{f}_{2}}~\left( 3 \right)~=\text{ }b\]

\[{{f}_{2}}~\left( 4 \right)~=\text{ }c\]

⇒ Every element of A has different images in B.

So, \[{{f}_{2}}\] is one-one.

Surjectivity:

Co-domain of \[{{f}_{2}}\] = {a, b, c}

Range of \[{{f}_{2}}\] = set of images = {a, b, c}

⇒ Co-domain = range

So, \[{{f}_{2}}\] is onto.