Bijective Function

One of the fundamental concepts in mathematics, particularly with regard to relations and mapping, is the concept of bijective functions. A function is said to be bijective if it is both surjective (or onto) and injective (or one-to-one). By guaranteeing that every element in the domain maps uniquely to an element in the codomain and vice versa, these features make it possible for the elements of two sets to be precisely matched.

1.0Properties of Bijective Functions

A function f: A → B is considered bijective if every element in set B corresponds uniquely to an element in set A, and vice versa. This means that for every b ∈ B, there exists a unique a ∈ A such that f (a) = b. Here, b is referred to as the image of a, while a is the preimage of b.

The key characteristics of bijective functions can be outlined as follows:

1. Each element in the domain A must be associated with exactly one element in the codomain B.
2. No single element in A can correspond to multiple elements in B.
3. Every element in B must be linked to at least one element in A.
4. Similarly, no element in B can be connected to more than one element in A.

This one-to-one mapping ensures that all elements in both sets are perfectly paired without overlap or omission.

2.0Differences: Injective, Surjective & Bijective Functions

To grasp the idea of bijective functions, it's essential to understand injective, surjective, bijective functions:

3.0Demonstrating that a Function is Bijective

To establish that a function f: A → B is bijective, you must show that it satisfies both injectivity and surjectivity. Start by defining the mapping f, ensuring that every element in A has a corresponding element in B.

• Injective Proof: Demonstrate that f (a₁) = f (a₂)  implies a₁ = a₂​. This ensures no two elements from the domain are mapped to the same element in the codomain.
• Surjective Proof: Confirm that every element in B has at least one preimage in A.

If the sets A and B differ in size (∣A∣ ≠ ∣B∣), a bijective mapping is impossible. When ∣A∣ = ∣B∣ = n, n!  unique bijections exist, pairing elements from A to B in all possible one-to-one correspondences.

This approach highlights bijections as precise pairings between sets.

4.0Solved Problems

Problem 1: Prove f (x) = 2x + 3 is Bijective

1. Injectivity: Suppose f (x₁) = f (x₂)
   2x₁ + 3 = 2x₂+ 3  ⟹  x₁ = x₂ 

Thus, f (x) is injective.

2. Surjectivity: Let y ∈ R, then solve y = 2_x + 3y for x
   x = y−3/2

Since x ∈ R, every y has a corresponding x. Hence, f (x) is surjective.

Therefore, f (x) = 2x + 3 is bijective.

Problem 2: Inverse of a Bijective Function

Solution: Find the inverse of f (x) =

$\frac{x-1}{x+2}$

Let y =

$\frac{x-1}{x+2}$​, then rearrange to solve for x:
y (x+2) = x − 1  ⟹  xy +2y = x−1 

x (y−1) = −1 −2y  ⟹  

$x=\frac{-1-2y}{y-1}$

Thus, the inverse function is:
f⁻¹ (y) =

$\frac{-1-2y}{y-1}$

Problem 3: 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?

• 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.

5.0Applications of Bijective Functions

Bijective functions have broad applications in various fields:

• Cryptography: Secure encoding and decoding use bijections to ensure unique mappings.
• Set Theory: Establishing equivalences between infinite sets often relies on bijective mappings.
• Data Structures: Hash functions aim to mimic bijective behaviour for efficient data retrieval.