Bijective Function (Bijection)

A bijective function creates a perfect one-to-one match between two sets, such as set A and set B. If a function is said to be bijective, then it should satisfy the following two properties:

Injectivity: Each element from set A must be connected with a unique element in set B. In simple terms, no two elements from set A can connect with the same element in set B.
Surjectivity: Every element in set B must be the image of at least one element in set A. This function should map every element of set B to at least one element in set A, meaning all elements of set B are included. 

When a function satisfies these two properties, such as injectivity and surjectivity, then it is called a bijective function. A bijective function establishes a perfect one-to-one correspondence between two sets. 
 

In mathematics, a function links each input to an output. The way inputs and outputs are paired can be different. Based on this pairing, functions are classified as injective, surjective, or bijective. Let’s look at the differences between these functions.

The following image shows the difference between surjective, injective, and bijective functions:

What are the Properties of Bijective Function?

The main properties of a bijective function are injective and surjective, but other than that, some more properties of bijective functions are:

• Inverse Exists
• Unique Inverse
• Preservation of Composition

Inverse Exists: The Inverse of a bijective function exists because the function pairs each element of the domain with a unique element of the codomain. The inverse reverses this mapping, taking an element from the codomain back to its original element in the domain.

Unique Inverse: The inverse of a bijective function is always unique, meaning there is only one function that can reverse the mapping.

Preservation of Composition: If two functions are bijective, then their composition is also bijective. 
 

For identifying a bijective function, we use two main processes, which are:

• Injectivity
• Surjectivity

Step 1: Check for Injectivity
Imagine two boxes, Box A and Box B.
Box A has: {1, 2, 3}
Box B has: {a, b, c}
We can use a function f to connect items from Box A to Box B,
f(1) = a
f(2) = b
f(3) = c
Now, check whether each number from Box A is connected with a different letter in Box B.
1 goes to a
2 goes to b
3 goes to c
Here, no two numbers go to the same letter. So it is injective, that is, one-to-one. 

Step 2: Check for Surjectivity
Next, see every letter in Box B is being used.
a is used by 1
b is used by 2
c is used by 3
No letter is left out. Each element in Box B is connected to something in Box A. 
This means the function is surjective; every element in Box B is covered.

If a function passes both injective(one-to-one) and surjective(onto), then the function is said to be bijective. 

How to Represent Bijective Function in Graph?

Let us consider a bijective function as f(x) = x. It is a linear function with a slope that is equal to 1. Let us see it clearly through a graph given below:
 

In real life, a bijective function is useful in many areas like technology, coding, mathematics, and even everyday tasks. Given below are some of the real-life applications of a bijective function.

• Cryptography: When we send a secret message using a code, a bijective function helps to convert each letter or number into a unique symbol or value. Since it is a bijective function, the original message can be perfectly reversed using the inverse function, without losing or mixing up any data.
• Computer Programming: In programming, bijective functions are used when each memory address must store exactly one value, and each value must have a unique memory address. This one-to-one matching prevents duplication or loss of data.
• Database Management: In schools, banks, or hospitals, each person is given a unique ID number. A bijective function links one person to one ID, and vice versa. 
• Matching Games or Puzzles: In games like memory cards or matching puzzles, each card is paired with exactly one other. A bijective function can be used to design or solve these games by creating a one-to-one link between the cards.
• Barcode Systems: Each product has a unique barcode, and the barcode scanner uses a function to link the code to the product. This function is bijective, so that no two products share the same code and every code identifies one product only.