Into Function and Bijective function - Practice Questions & MCQ

Into Function:

A function $f: X \rightarrow Y$ is said to be an into function if there exists an element in $Y$ having no preimage in A .

In other words, if $\mathrm{f}: \mathrm{X} \rightarrow \mathrm{Y}$ is not onto mapping then it is an into mapping.

Eg

As the element y₂ in codomain does not have a pre-image in domain, so it is into function

Note: If a function is not onto, then it is into and

If a function is not into, then it is onto.

Bijective Function

A function $f: X \rightarrow Y$ is said to be bijective, if $f$ is both one-one and onto (meaning it is both injective and surjective)

Consider, $X_1=\{1,2,3\}$ and $X_2=\{x, y, z\}$
Eg
$\mathrm{f}: \mathrm{x}_1 \rightarrow \mathrm{x}_2$

The number of bijective function: 

If $f(x)$ is bijective, and the function is from a finite set $A$ to a finite set $B$, then

$
n(A)=n(B)=m(\text { Say })
$
And, the number of Bijective functions $=\mathrm{m}$ !

Equal Functions  

The two functions $f$ and $g$ are said to be equal if
$\operatorname{Domain}(\mathrm{f})=$ Domain(g)
Co-domain(f) = Co-domain(g), and
$f(x)=g(x)$ for all $x$ belonging to domain