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:
- 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.
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:
- 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) = ex, where f : R → (0, ∞)
- Absolute Value Function: f(x) = x|x| (bijective only when the domain is restricted to x ≥ 0 or x ≤ 0)
Properties of Bijective Function
Other than surjectivity 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 co-domain 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 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:
- 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.
![Graph of Bijective Function [f(x) = x]](/https://media.geeksforgeeks.org/wp-content/uploads/20231019170640/Bijective-Function-1.png)
Injective vs Surjective vs Bijective Function
Following illustration shows the difference between all three 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 | Multiple x can map to the same f(x) | Each x1 |
| 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 |
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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) = 104In 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 x = (y-1)/2 . 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?