One to One Function - Practice Questions & MCQ

A function $f: X \rightarrow Y$ is called a one-one (or injective) function, if different elements of $X$ have different images in B. i.e. no two elements of set $X$ can have the same image.

Consider,
$f: X \rightarrow Y$, function given by $y=f(x)=x$, and
$X=\{-2,2,4,6\}$ and $Y=\{-2,2,4,6\}$,
Graphically it can be shown that for every x , there is a unique y (or no y has more than one x corresponding to it) as below and hence it is one-one.

Now, consider, $\mathrm{X} 1=\{1,2,3\}$ and $\mathrm{X} 2=\{\mathrm{x}, \mathrm{y}, \mathrm{z}\}$

$
\mathrm{f}: \mathrm{X} 1 \longrightarrow \mathrm{X} 2
$

Method to check One-One Function

If $\mathrm{x}_1, \mathrm{x}_2 \in \mathrm{X}$, then $\mathrm{f}\left(\mathrm{x}_1\right)=\mathrm{f}\left(\mathrm{x}_2\right) \Rightarrow \mathrm{x}_1=\mathrm{x}_2$
A function is one - one iff no line parallel to the X -axis meets the graph of the function at more than one point.

Even degree polynomials are NOT one-one functions

Number of One-One Function

If A and B are finite sets having elements m and n respectively, then the number of one-one functions from A to B is 

$=\left\{\begin{array}{cl}{ }^n P_m & \text { if } n \geq m \\ 0 & \text { if } n<m\end{array}\right.$