TITLE: What's the difference between "relation", "mapping", and "function"?
QUESTION [19 upvotes]: I think that a mapping and function are the same; there's only a difference between a mapping and relation. But I'm confused. What's the difference between a relation and a mapping and a function?

REPLY [6 votes]: In Mathspeak:
The words 'function' and 'mapping' are synonymous in most contexts. Formally, the definition of a function $f$ requires that for each value $x$ in the domain, there exists one and only one value $y$ in the codomain such that $y=f(x)$*.
A relation is any assignment of values in one set to values in another, regardless of whether or not the assignment is unique. For example, if $(X,Y,R)$ is the relation from a domain $X\subseteq\mathbb{R}$ to a range $Y\subseteq\mathbb{R}$ such that $xRy\iff x=y^2$, then the set of all points $\left\{(x,y)\in\mathbb{R}^2\mid x=y^2\right\}$ is a relation, but not a function, because there are two values $y\in Y$ such that $y^2=x$ for each $x\in X$.
It is worth pointing out that while every function is a relation, not every relation is a function. As stated by Wuestenfux, a function is the special case of a relation which is both left-total and right-unique.
In English:
A function is a relation where each input corresponds to a single output. A relation with more than one output for each input is still a relation, but it is not a funciton. For example, $y=\pm\sqrt{x}$ is a relation, because it relates $x$ to $y$, but not a function, because there are two values ($\sqrt{x}$ and $-\sqrt{x}$) for each input $x$.

In regards to user2662833's answer:
I believe that what user2662833 is trying to say is that while not every relation is a function, a relation over $n$ variables which is not a function, contains the inputs to a function of $n$ variables for which that function is constant. In the example given the points $(x,y)$ specified by the equation $x^2+y^2=4$ are the points for which $f(x,y)=x^2+y^2-4$ is $0$, and $g(x,y)=x^2+y^2$ is 4.
With an additional restriction, this could be restated as "every relation of $n$ variables contains all zeros of a function of $n$ variables".

This should still be the case outside the real numbers.

*In complex analysis, specifically, this is not always required as some authors prefer to regard certain complex 'multivalued functions' as functions.