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Produced by Charles Wells     Revised 2015-10-06
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Properties of functions

Injectivity

Definition

A function $F:A\to B$ is injective if for any elements $a,\,\,a'$ in the domain $A$,

If $a\ne a'$ then $F(a)\ne F(a')$.

It is the whole function that is or is not injective.

Useful rewordings of the definition:

Usage

To say "$F$ is injective", you can also say

Warning:  Do not try to reword this definition using the word "unique". It is too easy to get it mixed up with the definition of functional property. This is explained in the article on dysfunction.

Examples

See Wikipedia for more examples and more discussion.

Horizontal line crosses the graph only once at most

Let $F:\mathbb{R}\to \mathbb{R}$ be a real continuous function.  Then $F$ is injective if no horizontal line cuts the graph of the function twice.  This is a useful way of thinking about injective continuous functions, but it doesn’t work with arbitrary functions.

Examples

No information loss

An injective function $F:A\to B$ loses no information.  If you have an output from the function, you know it came from exactly one input.

Surjectivity

Definition

A function $F:A\to B$ is surjective if and only if for every element $b$ in the codomain B there is an element $a$ in the domain $A$ for which $F(a)=b$.

Useful rewordings of the definition:

Examples

Usage

Another way of saying that $F:A\to B$ is surjective is to say "$F$ is onto" or "$F$ is onto $B$."

Every horizontal line crosses the graph at least once.

A function $F:\mathbb{R}\to \mathbb{R}$ is surjective if every horizontal line crosses its graph one or more times. The graphs above suggest (correctly) that $x\mapsto x^3:\mathbb{R}\to \mathbb{R}$ and $x\mapsto x^3-x:\mathbb{R}\to \mathbb{R}$ are surjective, but $x\mapsto x^2:\mathbb{R}\to \mathbb{R}$ and $x\mapsto \sin x:\mathbb{R}\to \mathbb{R}$ are not surjective.

However, $x\mapsto x^2:\mathbb{R}\to \mathbb{R}^+$ and $x\mapsto \sin x:\mathbb{R}\to [0,1]$ are surjective. ($[0,1]$ is the closed interval $\{x\in\mathbb{R}|-1\lt x\lt 1\}$.)

Bijectivity

Definition

A function that is both injective and surjective is said to be bijective.

Usage

A bijection $F:A\to B$ matches up the elements of $A$ and $B$ -- each element of $A$ corresponds to exactly one element of $B$ and each element of $B$ corresponds to exactly one element of $A$. For this reason, a bijection is also called a one-to-one correspondence.

Examples

Properties of finite functions

You can see whether a finite function is injective, surjective or both by looking at its cograph.

Examples


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