Injective Functions (One-to-One Functions) in Discrete Mathematics

Understand injective functions, functions where each input maps to a unique output. This guide defines injective functions, explains how to determine if a function is injective using the horizontal line test, and provides examples to illustrate this key concept in discrete mathematics.

Injective Functions in Discrete Mathematics

Introduction to Injective Functions

An injective function, also known as a one-to-one function, is a type of function where each element in the domain maps to a unique element in the codomain. In simpler terms, no two different inputs produce the same output. This is a fundamental concept in mathematics and has applications in various areas.

Definition of an Injective Function

Let's say we have a function f with a domain A and a codomain B. The function f is injective if, and only if, for every a and b in A, if f(a) = f(b), then a = b. Another way to state this is that if a ≠ b, then f(a) ≠ f(b). Injective functions never map two different inputs to the same output.

Visualizing Injective Functions: The Horizontal Line Test

A graphical way to check if a function is injective is the horizontal line test. If any horizontal line intersects the graph of the function more than once, the function is *not* injective. If every horizontal line intersects the graph at most once, the function is injective.

(A diagram showing examples of injective and non-injective functions and the horizontal line test would be very useful here.)

Examples of Injective Functions

Example 1: f(a) = 2a

(Show how this function maps different inputs to different outputs, demonstrating it's injective.)

Example 2: f(a) = a/2

(Show how this function maps different inputs to different outputs, demonstrating it's injective.)

Example 3: f(a) = a²

(Show how this function maps both positive and negative values of 'a' to the same output values, demonstrating that it is *not* injective.)

Example 4: f(a) = a³

(Show how this function maps different inputs to different outputs, demonstrating it's injective.)

Injective Functions and Inverses

Only injective functions have inverses. If a function f is injective, its inverse function, denoted as f⁻¹, exists. The inverse function "undoes" the original function: f⁻¹(f(x)) = x and f(f⁻¹(y)) = y.

Example: Finding the Inverse of an Injective Function

(An example demonstrating how to find the inverse of a given injective function should be included here.)

Properties of Injective Functions

  • The decimal representation of an irrational number is non-terminating and non-repeating.
  • Irrational numbers are a subset of real numbers.
  • The sum of a rational and an irrational number is always irrational.
  • The product of a non-zero rational number and an irrational number is always irrational.
  • The least common multiple (LCM) of two irrational numbers may or may not exist.
  • The result of arithmetic operations on two irrational numbers may be rational or irrational.

Injective functions are a fundamental concept in mathematics with broad applications. Understanding their properties, how to identify them, and how to determine their inverses is essential for working with mathematical functions and various related areas.

Injective Functions: Understanding One-to-One Mappings

What is an Injective Function?

An injective function, also called a one-to-one function, is a type of function where every element in the domain (the set of input values) maps to a unique element in the codomain (the set of possible output values). In simpler terms, no two different inputs produce the same output. If f(a) = f(b), then a = b.

Properties of Injective Functions

  • The domain and range have the same cardinality (number of elements).
  • An injective function satisfies the properties of a function (which means there is only one output for each input).
  • The graph of an injective function will pass the horizontal line test (any horizontal line will intersect the graph at most once).

Examples of Injective Functions

Example 1: Determining if a Relation is Injective

Let X = {1, 2, 3} and Y = {u, x, y, z}. Determine which of these relations is an injective function:

  1. {(1, z), (2, z), (3, z)}
  2. {(1, x), (2, y), (3, z)}
  3. {(1, y), (1, z)}

Solution: Option 2 is the only injective function because each element in set X is mapped to a unique element in set Y.

Example 2: f(x) = 3x³ - 4

Show whether the function f(x) = 3x³ - 4 is injective (where x ∈ ℝ).

Solution: Assume f(x₁) = f(x₂). This leads to x₁³ = x₂³, which implies x₁ = x₂. Therefore, the function is injective.

Example 3: Composite Function gof(x)

Given f(x) = x + 1 and g(x) = 2x + 3, find g o f(x) and determine if it's injective.

Solution: g o f(x) = g(f(x)) = 2(x + 1) + 3 = 2x + 5. This function is injective because each input produces a unique output.

Example 4: f(a) = a³

Show whether f(a) = a³ is injective (where a ∈ ℝ).

Solution: This function is injective because each input maps to a unique output.

The Horizontal Line Test

The horizontal line test provides a visual way to determine if a function is injective. If any horizontal line intersects the graph of the function more than once, the function is *not* injective. If every horizontal line intersects the graph at most once, then the function is injective.

Parabolas and Injective Functions

The function f(x) = x² (a parabola) is *not* injective because it fails the horizontal line test (a horizontal line intersects the parabola twice for all positive outputs).

Injective Functions and Inverses

Only injective functions have inverse functions. If y = f(x), the inverse function, f⁻¹(y) = x exists if and only if f(x) is injective.

Example: Finding an Inverse Function

(An example finding the inverse of an injective function would be placed here.)

Conclusion

Injective functions are a fundamental concept in mathematics, providing a specific type of mapping between sets. Understanding their properties is crucial for various mathematical and computational tasks.