Into Function and Bijective Function: Definition & Differences

Functions can be defined as the relation between two sets where every element in one set has a unique element in another. A bijective functions are one of an important topic in set theory and Mathematics. These concepts are used in various fields like calculus, physics, engineering etc.

In this article, we will cover the concepts of bijective function. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of seven questions have been asked on this concept, including one in 2019, one in 2021, one in 2022, and four in 2023.

Function

$A$ and $B$ are two non-empty sets, then a relation from $A$ to $B$ is said to be a function if each element $x$ in $A$ is assigned a unique element $f(x)$ in $B$, and it is written as

$f: A ➝ B$ and read as $f$ is mapping from $ A$ to $B.$

Domain of a function

All possible values of $x$ for $f(x)$ is defined $(f(x)$ is a real number) is known as a domain.
If a function is defined from $A$ to $B$ i.e. f: $A \rightarrow B$, then all the elements of $\operatorname{set} A$ is called the Domain of the function.

Co-domain of a function

If a function is defined from $A$ to $B$ i.e. $f: A \rightarrow B$, then set $B$ is called the Co-domain of the function.

Range of a function: it is defined as all the values that the function assumes or in other words we can also say the output of the given function. It is also known as the image set of the function.

Injective function(one-one function)

A function $f: A \rightarrow B$ is called one - one function if distinct elements of $A$ have distinct images in $B$.

Eg. $f: X \longrightarrow Y$, function given by $y=f(x)=x$, and
$X=\{-2,2,4,6\}$ and $Y=\{-2,2,4,6\}$

Surjective Function(onto function)

A function $f: A \rightarrow B$ is said to be onto function if the range of $f$ is equal to the co-domain of $f$.

Eg.

$\mathrm{X}=\left\{\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4\right\}$ and $\mathrm{Y}=\left\{\mathrm{y}_1, \mathrm{y}_2, \mathrm{y}_3\right\}$

$
f: X \rightarrow Y
$

Bijective Function

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

Eg.
Consider, $\mathrm{X}_1=\{1,2,3\}$ and $\mathrm{X}_2=\{\mathrm{x}, \mathrm{y}, \mathrm{z}\}$

$
f: X_1 \rightarrow X_2
$

The function f is a injective function as the every distinct elements in $X_1$ has unique images in $X_2$ and a surjective function as every elemt in $X_2$ has a pre-image in $X_1$.

The number of bijective functions:

If $\mathrm{f}(\mathrm{x})$ is bijective, and the function is from a finite set A to a finite set B , then $n(A)=n(B)=m($ Say $)$ And, the number of Bijective functions $=\mathrm{m}$ !

Solved Examples Based On the Into and Bijective Functions:

Example 1: Let $f: N \rightarrow Y$ be a function defined as $f(x)=4 x+3$ where $Y=\{y \in N: y=4 x+3$ for some $x \in y\}$. Show that $f$ is invertible and its inverse is?

Solution:
As we learned in
Bijective Function -
The function that is both one-on-one and onto is the Bijective Function.

$
f(x)=4 x+3
$

Let $y=4 x+3$

$
\frac{y-3}{4}=x=g(y)
$

Example 2: If $n(A)=3$ and $n(B)=5$. How many bijective functions are possible from $A$ to $B$ ?
Solution:

If $n(A)=n(B)=m$, then the number of bijective functions from $A$ to $B=m!$.
Here, $n(A) \neq n(B)$
So, bijective functions are not possible.
Hence, the answer is 0 .

Example 3: Which of the following function is a bijective function?

1) $f(x)=\frac{x^2-4}{x-2}$

2) $f(x)=x^2$

3) $f_3(x)=3 x+4$

4) $f_4(x)=\frac{x^2-1}{x+1}$

Solution:

$f_1(x)$ is not defined for $\mathrm{x}=2$, and it does not have 4 in its range, so into and thus not bijective $f_4(x)$ is not defined for $\mathrm{x}=-1$, and does not have -2 in its range, so not bijective $f_3(x)=3 \mathrm{x}+4$ is defined for $\mathrm{x} \in \mathrm{R}$ and it is a one-one and onto function: bijective function $f_2(x)$ is not a one - one function

Hence, the answer is the option 3.

Example 4: If $n(A)=4$ and $n(B)=4$. Then how many bijective functions can be formed from $A$ to $B$ ?
1) $24$
2) $6$
3) $120$
4) $0$

Solution:

As we have learned
Number of Bijective Function
If $\mathrm{f}(\mathrm{x})$ is a bijection, then $n(A)=n(B)=m($ Say $)$
And the number of Bijective functions $=m$ !
Here, number of bijective function $=4!=24$
Hence, the answer is the option 1.

Example 5: If set $A$ has $5$ elements and set $B$ has $3$ elements. How many bijective function can be formed from $A$ to $B$ ?

1) $120x$
2) $6$
3) $0$
4) $24$

Solution:

As we have learned

Number of Bijective Function
If $\mathrm{f}(\mathrm{x})$ is bijective then $n(A)=n(B)=m($ Say $)$
And the number of Bijective functions $=m$ !

Here

There can be no bijective function from $A$ to $B$ since number of elements should be same for both the sets to form a bijective function

Hence, the answer is the option 3.

Summary

An into function is a type of function where not all elements of the codomain are mapped to by elements in the domain. This results in the range being a proper subset of the codomain. Understanding into functions is crucial for various mathematical theories, computer science applications, and real-world modeling scenarios where not all outcomes are possible or achievable. Recognizing and determining into functions helps in understanding the limitations and behavior of mappings between sets.