Bijective Function

Introduction

A bijective function is a one-one and onto function. In this tutorial, we will learn about functions, and their properties, such as injectivity and surjectivity. We will also learn about bijective functions and the invertibility of a function.

A function is defined as a mapping between two sets, the first set is called the domain, and the second set is called the co-domain if the elements of the domain have a unique image in the codomain.

A function is said to be injective or one-one if all the elements of the domain are mapped to a unique element in the co-domain. And a function is said to be surjective or onto if all the elements of the co-domain are mapped from some element of the domain. And if a function is both injective and surjective, then it is called bijective. A bijective function is also invertible.

Functions

A function is defined as a mapping between two sets, say A and B, such that every element of A has a unique image in B, i.e., no two elements in B are mapped from the same element of A.

$$\mathrm{f\:\colon\:a\:\rightarrow\:B\:\:\:\:\:such\:that\:f(x)\:=\:y}$$

$$\mathrm{Where\:,\:x\;\varepsilon\:A\:,\:y\varepsilon\:B}$$

Injectivity (One-One)

Injectivity of a function is defined as the property of the function such that no two elements of A are mapped to the same element of B i.e. every element of B has a unique pre-image.

Algebraically, the injectivity of a function 𝑓 is verified by a simple method given below. Let $\mathrm{f\:\colon\:A\rightarrow\:B}$ be a function,

Firstly, we will try to come up with a counter-example of two elements in the domain such that they map to the same element in the co-domain. If there exists no trivial counter-example,

Then let, $\mathrm{x_{1}\:,\:x_{2}\:\varepsilon\:A}$ be two elements in the domain such that they map on the same image in B,

$$\mathrm{i.e,,f(x_{1})\:=\:f(x_{2})}$$

Now, we will simplify the equation and try to separate the variables on both sides.

If the equation can be simplified into, $\mathrm{x_{1}\:=\:x_{2}}$, then the function 𝑓 is injective, but if there is any other condition that can be true, then the function 𝑓 is many-one or simply put, not injective.

Surjectivity (Onto)

Surjectivity of a function is defined as the property of the function such that all the elements of B are mapped from some element of A, i.e., all the elements of B have a pre- image in A.

It can also be defined as when the range of the function is equal to its co-domain.

Algebraically, the surjectivity of the function 𝑓 can be determined by the following simple steps,

Let $\mathrm{f\:\colon\:A\rightarrow\:B;\:f(x)\:=\:y}$

Firstly, we will try to find a counter-example, where for some $\mathrm{y\varepsilon\:B}$, there is no $\mathrm{x\varepsilon\:A}$, such that $\mathrm{f(x)\:=\:y}$. If there exists no trivial counter-example,

Then we will put the expression for the function in the equation $\mathrm{f(x)\:=\:y}$ and separate 𝑥 from the rest of the equation.

This will result in a function, say 𝑔, such that

$$\mathrm{g\colon\:B\:\rightarrow\:A;\:g(y)\:=\:x}$$

Then, if $\mathrm{x\:=\:g(x)\:\varepsilon\:A\:,\:∀\:y\:\varepsilon\:B }$ then the function 𝑓 is surjective. Otherwise, the function 𝑓 is into, or simply put, not surjective.

Bijective Functions

When a function is both injective and surjective, it is said to be a bijective function.

Invertibility

The invertibility of a function is defined by the operation of the composition of functions.

The invertibility of a function depends on its bijectivity, i.e., if a function is bijective, then it is also invertible.

The inverse function $\mathrm{f^{-1}}$ of a function 𝑓, defined by $\mathrm{f\colon\:A\rightarrow\:B;\:\:f(A)\:=\:B}$, is defined as,

$$\mathrm{f^{-1}\:\colon\:B\rightarrow\:A;\:f^{-1}(B)\:=\:A}$$

And, let I be the identity function, defined as $\mathrm{I\colon\:A\rightarrow\:A;\:\:I(x)\:=\:x\:,\:∀\:x\varepsilon\:A}$

Then,

$$\mathrm{fof^{-1}(B)\:=\:B\:and\:f^{-1}of\:(A)\:=\:A}$$

Solved Examples

1) Let 𝒇 be a function defined as, $\mathrm{f\colon\:N\rightarrow\:N;\:f(x)\:=\:x^{2}}$. Check if 𝒇 is bijective. Also, if 𝒇 is bijective find its inverse.

Answer − Injective −

Let, $\mathrm{x_{1}\:,\:x_{2}\:\varepsilon\:N}$

Such that, $\mathrm{f(x_{1})\:=\:f(x_{2})}$

$$\mathrm{\Longrightarrow\:{x_{1}}^{2}\:=\:{x_{2}}^{2}}$$

$$\mathrm{\Longrightarrow\:x_{1}\:=\:\pm\:x_{2}}$$

Since, the set is of Natural numbers, there are no negative numbers,

$$\mathrm{\Longrightarrow\:x_{1}\:=\:x_{2}}$$

Hence, the function is injective. Surjective −

We have a counter-example, Let $\mathrm{y\:=\:2\:\varepsilon\:N}$ (codomain) There doesn’t exists any $\mathrm{x\varepsilon\:N}$ (domain) Such that, $\mathrm{y\:=\:f(x)\:=\:x^{2}}$

Hence, the function is not surjective and by definition not bijective.

Conclusion

A function is defined as a mapping between two sets, the first set is called the domain, and the second set is called the co-domain, if the elements of the domain have a unique image in codomain. A function is said to be injective or one-one if all the elements of domain are mapped to a unique element in co-domain. And a function is said to be surjective or onto, if all the elements of co-domain are mapped from some element of the domain. And if a function is both injective and surjective then it is called bijective. A bijective function is also invertible.

FAQs

1. What is a function?

A function is defined as a mapping between two sets, say A and B, such that every element of A has a unique image in B, i.e. no two element in B are mapped from the same element of A.

$$\mathrm{f\colon\:A\rightarrow\:B\:\:\:\:\:such\:that\:f(x)\:=\:y}$$

$$\mathrm{Where\:,\:x\:\varepsilon\:A\:,\&\:y\:\varepsilon\:B}$$

2. What is an injective function?

A function is said to be injective if the every element of the domain is mapped to a unique element of the co-domain.

3. What is a surjective function?

A function is said to be surjective if the range of the function is equal to its co- domain.

4. What do you mean by a bijective function?

When a function is both injective and surjective it is said to be a bijective function.

5. Define the inverse of a function?

The inverse function 𝑓−1 of a function 𝑓, defined by $\mathrm{f\colon\:A\:\rightarrow\:B;\:\:f(A)\:=\:B}$, is defined as,

$$\mathrm{f^{-1}\colon\:B\rightarrow\:A;\:f^{-1}(B)\:=\:A}$$

and, let I be the identity function, defined as $\mathrm{fof^{-1}(B)\:\:=\:B\:and\:f^{-1}of(A)\:=\:A}$ then,

$$\mathrm{}$$

6. How to verify injectivity and surjectivity of a function?

Injectivity − The injectivity of a function 𝑓 is verified by a simple method given below. Let $\mathrm{f\colon\:A\rightarrow\:B}$ be a function,

Firstly, we will try to come up with a counter example of two elements in domain such that they map to the same element in the co-domain. If there exists no trivial counter- example,

Then let, $\mathrm{X_{1},X_{2}\varepsilon\:A}$ be two elements in the domain such that they map on the same image in B, i.e., 𝑓(𝑥1) = 𝑓(𝑥2)

Now, we will simplify the equation and try to separate the variables on both sides. If the equation can be simplified into, $\mathrm{x_{!}\:=\:x_{2}}$, then the function 𝑓 is injective.

Surjectivity − The surjectivity of the function 𝑓 can be determined by the following simple steps,

Let $\mathrm{f\colon\:A\rightarrow\:B;\:f(x)\:=\:y}$

Firstly, we will try to find a counter-example, where for some $\mathrm{y\varepsilon\:B}$, there is no $\mathrm{x\varepsilon\:A}$, such that $\mathrm{f(x)\:=\:y}$. If there exists no trivial counter-example,

Then we will put the expression for the function in the equation and separate 𝑥 from the rest of the equation.

This will result in a function, say 𝑔, such that,

$$\mathrm{}$$

Then, if $\mathrm{x\:=\:g(x)\:\varepsilon\:A\:,\:∀\:y\:\varepsilon\:B}$ then the function 𝑓 is surjective.