One to One Function - Practice Questions & MCQ

A function $f:X\to Y$ 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,
$f:X\to Y$, 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 $\mathrm{X}2=\{\mathrm{x},\mathrm{y},\mathrm{z}\}$

$\mathrm{f}:\mathrm{X}1⟶\mathrm{X}2$

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 

$=\{\begin{array}{cl}{}^n{P}_m& \text{ if }n\ge m\\ 0& \text{ if }n<m\end{array}$