How to Prove a Function is Not Injective

Proving that a function is not injective is an important task in many areas of mathematics and computer science. This article will guide you through the process of demonstrating that a function f: A → B is not injective, based on the definition and steps involved. We will use a step-by-step approach to understand and apply the concept, along with an example to make it clearer.

Understanding Injectivity

A function f is said to be injective or one-to-one if for every pair x_1, x_2 in the domain A, whenever f(x_1) f(x_2), it necessarily follows that x_1 x_2. In simpler terms, different inputs must always produce different outputs. This property can be negated to prove that a function is not injective.

Finding a Counterexample

To show that a function f is not injective, you need to find a specific pair of distinct elements a_1, a_2 ∈ A such that:

a_1 ≠ a_2 f(a_1) f(a_2)

This pair of elements contradicts the definition of injectivity and thus proves that the function is not injective.

Constructing the Argument

The argument involves clearly stating the findings. Typically, this includes:

Identifying the distinct elements a_1, a_2 in the domain A such that a_1 ≠ a_2. Showing that f(a_1) f(a_2). Concluding that since a_1 ≠ a_2 and f(a_1) f(a_2), the function f is not injective.

Example: Proving f(x) x^2 is Not Injective

Consider the function f(x) x^2 defined on the real numbers mathbb{R}. To prove that this function is not injective, we need to find a specific pair of distinct real numbers a_1, a_2 ∈ mathbb{R} such that f(a_1) f(a_2).

Choose the values a_1 2 and a_2 -2. Compute f(a_1) and f(a_2) as follows: f(2) 2^2 4 f(-2) (-2)^2 4 Since f(2) f(-2) 4 and 2 ≠ -2, we conclude that f(x) x^2 is not injective.

Summary

The process of proving that a function is not injective involves finding distinct inputs that yield the same output, thereby serving as a counterexample to the injectivity condition. By systematically identifying such pairs and explicitly demonstrating them, you can effectively prove the non-injectivity of a given function.

Keywords: injective function, function not injective, counterexample