# difference between function and relation?

What is the difference between function and relations ? What are the characteristics of function and relations? What are the similarities and contrasts between relations and functions ? Examples will be much appreciated , thanks !

• In non formal terms: A function defines that one input only has ONE output. A relation does not require that. One input can have many outputs. So a (non vertical) line is a function, but a points on a circle form a relation Sep 17, 2016 at 15:18
• One could call relations "multi-valued functions". Sep 17, 2016 at 17:36
• Sometimes I like to think of a relation $R$ as a boolean function that takes two arguments $R(x,y)$ and outputs either $true$ or $false$ whether $(x,y)$ belongs to $R$ or not. For example say $R$ is the "$\leq$-function", then $R(2,3) = true$. Of course this is not the formal definition, just a way of thinking. Sep 17, 2016 at 17:39

Relation between sets $X$ and $Y$ is a triple $(X,Y,R)$ where $R$ is an arbitrary subset of $X\times Y$.

Example 1. Relation $\le$ between integers is $(\mathbb Z,\mathbb Z, R)$ where $(x,y)\in R$ if $x$ is less than or equal to $y$, e. g. $(2,3)\in R$, $(3,2)\notin R$. This is an example of a total order relation.

Example 2. Biological relation between animals $$\big(\mathrm{animals},~\mathrm{animals},~\{(x,y)\mid x,y\text{ are the same species}\}\big)$$ is an example of equivalence relation.

Function (map) is a relation $(X,Y,R)$ such that for every $x\in X$ there is unique element $(x,y)\in R$.

The keywords are "every" and "unique", this is the difference from general relations. $X$ is called domain, $Y$ is codomain, $\{y\mid (x,y)\in R\}$ is range, $R$ is graph of the function.

Example 3. Function $$f=\left(\mathbb R,\mathbb R,\{(x,y)\in \mathbb R^2 \mid y=x^2\}\right)\,,$$ is a parabola, usually denoted as $$f: \mathbb R\to \mathbb R,~ x\mapsto x^2\,.$$

Example 4. Function $$\big(\mathrm{animals},~\mathrm{species},~\{(x,y)\mid x\text{ belong the species } y\}\big)$$ is in fact the canonical projection of animals by equivalence relation from the example 2.

Use Google or Wikipedia for all unknown words.

• Not a mathematical comment, but a biological one: If one believes in evolution by natural selection, then "same species" is actually not an equivalence relation (not transitive, in a way that Really Forces One to Contemplate what transitivity means). Oct 7, 2022 at 16:47

In some interpretations, functions are relations with some very specific conditions. Specifically, a relation $R$ is a function is for any $x$ there is exactly one $y$ such that $xRy$.

Take, for instance, the relations on $\Bbb R$. There are a lot of relations. You might be aware of $\leq$, for instance, or the relation $\sim$ defined by $x \sim y$ iff $x-y \in \Bbb Z$.

Some of these relations are functions. For instance, the relation $R$ given by $xRy$ iff $x - y = 1$ is a function. It's usually written as $y = x-1$ or $R(x) = x-1$, but that doesn't change how it works.

'Relation' is one person to other person how to relate but function is transformation so function transform to one object to other object.

• this answer might be misguiding that relation is restricted to human relationship isn't it? Feb 21, 2018 at 2:54