According to the question,
![\[A\text{ }=\text{ }R\text{ }\text{ }\left\{ 3 \right\},\text{ }B\text{ }=\text{ }R\text{ }\text{ }\left\{ 1 \right\}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d8a42a661de6d5f9510ab3122c1e3895_l3.png "Rendered by QuickLaTeX.com")
And,
f : A → B be defined by
![\[f\text{ }\left( x \right)\text{ }=\text{ }x\text{ }\text{ }2/\text{ }x\text{ }\text{ }3\forall x\in A\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fd4b3f96519492b28df06c83716bcb6c_l3.png "Rendered by QuickLaTeX.com")
Hence,
![\[f\text{ }\left( x \right)\text{ }=\text{ }\left( x\text{ }\text{ }3\text{ }+\text{ }1 \right)/\text{ }\left( x\text{ }\text{ }3 \right)\text{ }=\text{ }1\text{ }+\text{ }1/\text{ }\left( x\text{ }\text{ }3 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d99026694455b5807cd839e10c8c32d6_l3.png "Rendered by QuickLaTeX.com")
Let
![\[f({{x}_{1}})\text{ }=\text{ }f\text{ }({{x}_{2}})\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d827fd9b142ea0af316ee2b2b9699195_l3.png "Rendered by QuickLaTeX.com")
So, f (x) is an injective function.
Now let
![\[y\text{ }=\text{ }\left( x\text{ }\text{ }2 \right)/\text{ }\left( x\text{ }-3 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c3622dd82d1775e173958ed6676a4e1f_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{array}{*{35}{l}} x\text{ }\text{ }2\text{ }=\text{ }xy\text{ }\text{ }3y \\ x\left( 1\text{ }\text{ }y \right)\text{ }=\text{ }2\text{ }\text{ }3y \\ x\text{ }=\text{ }\left( 3y\text{ }\text{ }2 \right)/\text{ }\left( y\text{ }\text{ }1 \right) \\ y\in R\text{ }\text{ }\left\{ 1 \right\}\text{ }=\text{ }B \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-072c3d58649f1834c25574b69e29d1ed_l3.png "Rendered by QuickLaTeX.com")
Thus, f (x) is onto or subjective.
Therefore, f(x) is a bijective function.
Let
![\[\mathbf{A}\text{ }=\text{ }\mathbf{R}\text{ }\text{ }\left\{ \mathbf{3} \right\},\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{R}\text{ }\text{ }\left\{ \mathbf{1} \right\}.\text{ }\mathbf{Let}\text{ }\mathbf{f}\text{ }:\text{ }\mathbf{A}\text{ }\to \text{ }\mathbf{B}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4826e8120534bd03f6ed6670ce6482c3_l3.png "Rendered by QuickLaTeX.com")
![\[\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{x}\text{ }\text{ }\mathbf{2}/\text{ }\mathbf{x}\text{ }\text{ }\mathbf{3}\forall \mathbf{x}\in \mathbf{A}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3be59825d072da5199347321d3e97a56_l3.png "Rendered by QuickLaTeX.com")
According to the question,
And,
f : A → B be defined by
Hence,
Let
So, f (x) is an injective function.
Now let
Thus, f (x) is onto or subjective.
Therefore, f(x) is a bijective function.