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Which of the following functions from A to B are one-one and onto?

    \[{{\mathbf{f}}_{\mathbf{3}}}~=\text{ }\left\{ \left( \mathbf{a},~\mathbf{x} \right),\text{ }\left( \mathbf{b},~\mathbf{x} \right),\text{ }\left( \mathbf{c},~\mathbf{z} \right),\text{ }\left( \mathbf{d},~\mathbf{z} \right) \right\};~\mathbf{A}~=\text{ }\left\{ \mathbf{a},~\mathbf{b},~\mathbf{c},~\mathbf{d}, \right\},~\mathbf{B}~=\text{ }\left\{ \mathbf{x},~\mathbf{y},~\mathbf{z} \right\}\]

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CBSE | Class 12 | Exercise 2.1 | Function | Maths | RD Sharma
Which of the following functions from A to B are one-one and onto?

    \[{{\mathbf{f}}_{\mathbf{3}}}~=\text{ }\left\{ \left( \mathbf{a},~\mathbf{x} \right),\text{ }\left( \mathbf{b},~\mathbf{x} \right),\text{ }\left( \mathbf{c},~\mathbf{z} \right),\text{ }\left( \mathbf{d},~\mathbf{z} \right) \right\};~\mathbf{A}~=\text{ }\left\{ \mathbf{a},~\mathbf{b},~\mathbf{c},~\mathbf{d}, \right\},~\mathbf{B}~=\text{ }\left\{ \mathbf{x},~\mathbf{y},~\mathbf{z} \right\}\]

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Consider

Injectivity:

    \[{{f}_{3}}~\left( a \right)\text{ }=\text{ }x\]

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

    \[{{f}_{3}}~\left( c \right)\text{ }=\text{ }z\]

    \[{{f}_{3}}~\left( d \right)\text{ }=\text{ }z\]

⇒ a and b have the same image x.

Also c and d have the same image z

So, 

    \[{{f}_{3}}~\]

 is not one-one.

Surjectivity:

Co-domain of 

    \[{{f}_{3}}~\]

 ={x, y, z}

Range of 

    \[{{f}_{3}}~\]

 =set of images = {x, z}

So, the co-domain  is not same as the range.

So, 

    \[{{f}_{3}}~\]

 is not onto.

i

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