Bijective Function (original) (raw)

Last Updated : 11 Nov, 2025

A bijective function also known as a 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:

Bijective 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.

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

Bijective Function Examples

Some examples of Bijective functions are:

Properties of Bijective Function

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

How to Identify a Bijective Function?

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

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:

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:

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.

Graph of Bijective Function [f(x) = x]

Injective vs Surjective vs Bijective Function

Following illustration shows the difference between all three function:

Injective, Surjective and Bijective Function

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

Property Injective Function Surjective Function Bijective Function
Injectivity Each x1 \neq x2 maps to distinct f(x) Multiple x can map to the same f(x) Each x1 \neq 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) = ex, for x ∈ R f(x) = x2 for for x ∈ R f(x) = x for x ∈ R

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) = 2__x+1. Is this function bijective?

**Solution:

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:

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

Practice Problems on Bijective Function

**Problem 1: Determine whether the following function is Bijective:

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