Bijective functions

Bijective Functions

A bijective function, also known as a bijection or one-to-one correspondence, is a function that is both injective (one-to-one) and surjective (onto). This type of function establishes a perfect pairing between the elements of its domain and codomain, meaning that every element of the domain maps to a unique element of the codomain, and every element of the codomain is mapped to by some element of the domain.

Definitions

Before diving into bijective functions, let's define some key terms:

  • Function (f): A relation from a set of inputs (domain) to a set of possible outputs (codomain) where each input is related to exactly one output.

  • Domain: The set of all possible inputs for a function.

  • Codomain: The set of all possible outputs for a function.

  • Injective (One-to-One): A function ( f: A \rightarrow B ) is injective if for every ( a_1, a_2 \in A ), ( f(a_1) = f(a_2) ) implies ( a_1 = a_2 ).

  • Surjective (Onto): A function ( f: A \rightarrow B ) is surjective if for every ( b \in B ), there exists at least one ( a \in A ) such that ( f(a) = b ).

  • Bijective (One-to-One Correspondence): A function is bijective if it is both injective and surjective.

Bijective Function Properties

A bijective function has the following properties:

  • Each element of the domain is paired with a unique element of the codomain.
  • Each element of the codomain has a pre-image in the domain.
  • A bijective function has an inverse function.

Table of Differences

Property Injective Surjective Bijective
Definition Each element of the domain maps to a unique element of the codomain. Every element of the codomain is mapped to by some element of the domain. Both injective and surjective properties hold.
Domain-Codomain Relationship One-to-one Onto One-to-one and onto
Inverse Not necessarily Not necessarily Always exists
Example ( f(x) = 2x ) for ( x \in \mathbb{R} ) ( f(x) = x^2 ) for ( x \in \mathbb{R} ) ( f(x) = x ) for ( x \in \mathbb{R} )

Formulas

For a function ( f: A \rightarrow B ), to prove that ( f ) is bijective, we must show two things:

  1. Injectivity: ( \forall a_1, a_2 \in A, f(a_1) = f(a_2) \Rightarrow a_1 = a_2 )
  2. Surjectivity: ( \forall b \in B, \exists a \in A \text{ such that } f(a) = b )

If a function ( f ) is bijective, then its inverse ( f^{-1} ) exists and satisfies the following:

( f^{-1}(f(a)) = a ) for all ( a \in A ) ( f(f^{-1}(b)) = b ) for all ( b \in B )

Examples

Example 1: Proving a Function is Bijective

Consider the function ( f: \mathbb{R} \rightarrow \mathbb{R} ) defined by ( f(x) = 2x + 3 ).

  • Injectivity: Assume ( f(x_1) = f(x_2) ). Then ( 2x_1 + 3 = 2x_2 + 3 ). Subtracting 3 from both sides and dividing by 2, we get ( x_1 = x_2 ), proving injectivity.

  • Surjectivity: For any ( y \in \mathbb{R} ), we can find an ( x \in \mathbb{R} ) such that ( f(x) = y ). Solving for ( x ), we get ( x = \frac{y - 3}{2} ), which is a real number. Therefore, the function is surjective.

Since the function is both injective and surjective, it is bijective.

Example 2: Finding the Inverse of a Bijective Function

Let ( f: \mathbb{R} \rightarrow \mathbb{R} ) be defined by ( f(x) = 3x - 4 ). To find the inverse, we solve for ( x ) in terms of ( y ):

( y = 3x - 4 ) ( x = \frac{y + 4}{3} )

The inverse function ( f^{-1}: \mathbb{R} \rightarrow \mathbb{R} ) is given by ( f^{-1}(y) = \frac{y + 4}{3} ).

Example 3: Non-Bijective Function

The function ( f: \mathbb{R} \rightarrow \mathbb{R} ) defined by ( f(x) = x^2 ) is not bijective:

  • It is not injective because ( f(1) = f(-1) = 1 ), but ( 1 \neq -1 ).
  • It is not surjective because there is no real number ( x ) such that ( f(x) = -1 ), as squares of real numbers are non-negative.

Understanding bijective functions is crucial for various fields of mathematics, including algebra, analysis, and combinatorics. They are fundamental in establishing isomorphisms between algebraic structures and in defining equivalences in set theory.