One-One Function

One-One Function

## Definition / Concept

A function is one-one or injective if distinct elements of the domain have distinct images.

## Formula

$$f(x_1)=f(x_2)\Rightarrow x_1=x_2$$

Equivalently:

$$x_1\ne x_2\Rightarrow f(x_1)\ne f(x_2)$$

## Conditions / Special Cases

For finite sets, if:

$$f:A\to B$$

is one-one, then:

$$n(A)\le n(B)$$

The number of one-one functions from an $m$-element set to an $n$-element set, where $n\ge m$, is:

$$\frac{n!}{(n-m)!}$$

## Usage

- Used to test injectivity algebraically or graphically.

- Used in counting problems involving one-one mappings.