Consider
![\[{{\mathbf{f}}_{\mathbf{2}}}~=\text{ }\left\{ \left( \mathbf{2},~\mathbf{a} \right),\text{ }\left( \mathbf{3},~\mathbf{b} \right),\text{ }\left( \mathbf{4},~\mathbf{c} \right) \right\};~\mathbf{A}~=\text{ }\left\{ \mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4} \right\},~\mathbf{B}~=\text{ }\left\{ \mathbf{a},~\mathbf{b},~\mathbf{c} \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1b3b79e1276e0e683d5ff5a717099856_l3.png "Rendered by QuickLaTeX.com")
Injectivity:
![\[{{f}_{2}}~\left( 2 \right)~=\text{ }a\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0c1781f7b18a5dd8d83e3a22ee4f8232_l3.png "Rendered by QuickLaTeX.com")
![\[{{f}_{2}}~\left( 3 \right)~=\text{ }b\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-61f990062a89365339a907b638752171_l3.png "Rendered by QuickLaTeX.com")
![\[{{f}_{2}}~\left( 4 \right)~=\text{ }c\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6cd8f07ec748a4d422e61b0c6f931919_l3.png "Rendered by QuickLaTeX.com")
⇒ Every element of A has different images in B.
So,
![\[{{f}_{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-53bf768d033827e60518a1eebe263199_l3.png "Rendered by QuickLaTeX.com")
is one-one.
Surjectivity:
Co-domain of
= {a, b, c}
Range of
= set of images = {a, b, c}
⇒ Co-domain = range
So,
is onto.