Bijective Function

Bijective Function is a special type of function that represents the relationship between two sets in such a way that all elements in the domain have an image in the codomain and each element in the codomain has a pre-image in the domain.

Bijective Function is also called one-to-one correspondence due to the relationship between domain and codomain i.e., each element of the domain is mapped to a unique element of codomain, and no element of codomain remains left without pre-image. In real life, the concept of bijective functions is used in various places such as shuffling of cards, cryptography, biometrics, physical lock and keys, language translation, etc. In this article, we have provided a well-detailed description of the concept of the Bijective Function, including all the subtopics.

What is Bijective Function?

A bijective function also known as bijection, ensures a perfect match between two sets, typically referred to as Set A and Set B. To be considered bijective, a function must satisfy these two properties:

• Injectivity: This means that each element from Set A must connect with a distinct element in Set B. In simpler terms, no two different elements from Set A can connect with the same element in Set B.
• Surjectivity: The function should cover the entire Set B. This means that for every element in Set B, there should be at least one element in Set A that connects with it through the function.

When a function satisfies both injectivity and surjectivity, it is classified as a bijective function, establishing a perfect one-to-one correspondence between the elements of Set A and Set B.

Bijective Function Definition

If function f satisfies all the conditions of injective and surjective functions, then function f is a bijective function.

In this figure, we can see that every element in the domain is connected with a corresponding element in the codomain for bijective function and not connected with unique element of . Therefore, it serves as an example of a bijective function.

Read more about Domain and Range.

Bijective Function Examples

Some examples of Bijective functions are:

• Linear Functions: f(x) = x, g(x) = x + 10, h(x) = 5x – 5, etc.
• Polynomial Functions: f(x) = x³, g(x) = x³ – 1 
• Exponential Functions: f(x) = e^x, where f : R → (0, ∞)
• Absolute Value Function: f(x) = x|x|

Properties of Bijective Function

Other than subjectivity and injectivity, there are some more properties of bijective function that are listed as follows:

• Inverse Exists: A bijective function has an inverse function that undoes the mapping, taking an element from the codomain back to an element in the domain.
• Unique Inverse: The inverse of a bijective function is unique, meaning there is only one function that reverses the mapping.
• Preservation of Composition: If you compose a bijective function with another function, the composition is also bijective.

How to Identify a Bijective Functions?

To figure out if a function is bijective, there is a 2 step process to identify:

1. Injectivity
2. Surjectivity

Let’s understand this step in brief.

Step 1: Check for Injectivity

Start by imagining you have two different sets: Set A and Set B. In Set A, we have some elements and in Set B, we have some other elements.

f(x₁) = f(x₂) ⇒ x₁ = x₂

Let’s consider and example to understand the check for injectivity,

Set A: {1, 2, 3}

Set B: {a, b, c}

Now, consider a function, which we’ll call f, that connects elements from Set A to Set B:

• f(1) = a
• f(2) = b
• f(3) = c

In the injectivity step, what you want to make sure is that no two different items from Set A (let’s say, x₁ and x₂) point to the same item in Set B through the function f. So, if f(x₁) = f(x₂), you need to prove that x₁ must be equal to x₂.

In our example, since each element from Set A maps to a distinct element in Set B, like 1 to a, 2 to b and 3 to c, there is no duplication. So, it passes the injectivity test.

Step 2: Check for Surjectivity

Next, we want to ensure that no element in Set B is left without a connection from Set A.

Example:

Set A: {1, 2, 3}

Set B: {a, b, c}

With the same f function:

• f(1) = a
• f(2) = b
• f(3) = c

In the surjectivity step, we confirm that for every item y in Set B (like a, b or c), there is a corresponding item x in Set A (1, 2 or 3) such that f(x) = y. This ensures that every item in Set B has a connection in Set A through the function f.

In our example, since each element in Set B (a, b and c) has a connection to an element in Set A (1, 2 and 3), the function f passes the surjectivity test.

When a function passes both the injectivity and surjectivity tests, like in our example, it’s classified as a bijective function. It establishes a one-to-one relationship between elements of Set A and Set B without any duplication or missing connections.

Graph of Bijective Function

As there are variuos different bijective functions, we can consider one such function i.e., f(x) = x. This is a linear function with slope equals to 1. Let’s see its graph as following illustration.

Injective, Surjective and Bijective Function

The key differences between Injective, Surjective and Bijective Function are listedin following table:

Property	Injective Function	Surjective Function	Bijective Function
Injectivity	Each x1 x2 maps to distinct f(x)	Multiple x can map to the same f(x)	Each x1 x2 maps to distinct f(x)
Surjectivity	Not necessarily every element in codomain is covered	Every element in codomain is covered but not necessarily distinctively	Every element in codomain is covered distinctly
Bijectivity	No	No	Yes
Symbol	↣	↠	⤖
Inverse Function	May not have an inverse	May not have an inverse	Has a unique inverse
Example	f(x) = 5x+5 for x ∈ R	f(x) = x3 for for x ∈ R	f(x) = x for x ∈ R

Following illustration shows the difference between all three function:

Read More,

• Relation and Function
• Types of Functions
• Trigonometric Functions

Solved Examples of Bijective Functions

Example 1: f(x) = x (Number to Itself)

Solution:

Let’s consider the set of natural numbers (positive whole numbers) as both Set A and Set B. In this case, you can define a bijective function like this:

f(1) = 1
f(2) = 2
f(3) = 3
…

In this function, every number in Set A (natural numbers) maps to the exact same number in Set B. It’s a straightforward example of a bijective function because it meets both criteria each element in Set A matches a distinct element in Set B (injectivity) and no elements in Set B are left unmatched (surjectivity).

Example 2: Matching Students and ID Numbers

Solution:

Suppose you have a set of students (Set A) and a set of student ID numbers (Set B). You want to create a bijective function to pair each student with a unique student ID number. Let’s say your sets look like this:

Set A (Students): {Alice, Bob, Carol, David}
Set B (Student ID Numbers): {101, 102, 103, 104}

You can define a bijective function f as follows:

f(Alice) = 101
f(Bob) = 102
f(Carol) = 103
f(David) = 104

In this example, each student in Set A is matched with a distinct student ID number in Set B, ensuring that no two students share the same ID number (injectivity). Also, every student ID number in Set B has a corresponding student in Set A, so no ID numbers are left unassigned (surjectivity).

Example 3: Consider the function f:R→R defined as f(x) = 2x+1. Is this function bijective?

Solution:

• Injectivity: To check injectivity, assume that f(x₁) = f(x₂). for two different real numbers x₁ and x₂. This leads to the equation 2x₁ + 1 = 2x₂ + 1. After some analysis, we find that x₁ must be equal to x₂. This proves that the function is injective.
• Surjectivity: To check surjectivity, we need to ensure that for any real number y, we can find a corresponding value of x such that 2x+1=y. Solving this equation, we get . This shows that the function covers all real numbers in its range, confirming its surjectivity.

With both injectivity and surjectivity confirmed, the function f is indeed bijective.

Example 4: Let g:{1,2,3}{a,b,c} be defined as g(1)=a, g(2)=b and g(3)=c. Is g a bijective function?

Solution:

• Injectivity: In this case, each element from the domain (1, 2 and 3) maps to a distinct element in the codomain (a, b and c). This means the function is injective.
• Surjectivity: The function covers all the elements in the codomain (a, b and c). Since every element in the codomain has a connection in the domain, the function is surjective.

Considering these factors, g unquestionably qualifies as a bijective function.

Practice Problems on Bijective Function

Problem 1: Determine whether the following function is Bijective:

• f(x) = 2x + 5
• g(x) = x² + 1
• k(x) = 5x – 2

Problem 2: Consider the function f(x) = 1/(x – 10) for x ≠ 10: Is p(x) an bijective function?

Bijective Functions: FAQs

1. Define Bijective Function.

A bijective function, also known as a bijection or one-to-one function, is a function that connects two sets, Set A and Set B. In this function, every element from Set A points to a distinct element in Set B and it covers the entire Set B.

2. How Do You Determine If a Function Is Bijective?

To determine if a function is bijective, we need to check two things:

• Injectivity: This means that no two different items from Set A map to the same item in Set B through the function.
• Surjectivity: This ensures that every item in Set B has a corresponding item in Set A through the function.

3. What are the Two Conditions for a Function to be Bijective?

For a function to be bijective, it must satisfy two conditions:

• It must be injective, meaning that no two different items from Set A map to the same item in Set B.
• It must be surjective, indicating that every item in Set B has a corresponding item in Set A.

4. What Is the Difference Between Injective, Surjective, and Bijective Functions?

Injective functions (one-to-one) have unique image for each element of the domain. Surjective functions (onto function) cover the entire codomain. Bijective functions are both injective and surjective, establishing a one-to-one correspondence.

5. Can All Functions Be Bijective?

No, not all functions can be bijective. A function can only be bijective if it is both injective (one-to-one) and surjective (onto). Many functions are not injective or surjective.

6. What is One to One Correspondence?

One-to-one correspondence, or a bijection, is a relationship between two sets where each element in one set is paired with exactly one element in the other set, without duplication or omission.

7. Can a Finite Set Have a Bijective Function with an Infinite Set?

No, a finite set cannot have a bijective function with an infinite set. A bijective function implies a one-to-one correspondence, and it’s not possible between finite and infinite sets.

8. Are Bijections Unique Between Two Sets?

No, bijections between two sets are not unique. Different bijections can exist between the same two sets, but they all establish a one-to-one correspondence.