Definition:Image (Relation Theory)/Mapping/Element

Definition
Let $f: S \to T$ be a mapping. Let $s \in S$.

The image of $s$ (under $f$) is defined as:


 * $\operatorname{Im} \left ({s}\right) = f \left ({s}\right) = \bigcup \left\{ {t \in T: \left({s, t}\right) \in f}\right\}$

That is, $f \left ({s}\right)$ is the element of the codomain of $f$ related to $s$ by $f$.

By the nature of a mapping, $f \left ({s}\right)$ is guaranteed to exist and to be unique for any given $s$ in the domain of $f$.

Also known as
This is also called the functional value, or value, of $f$ at $s$.

The terminology:
 * $f$ assigns the value $f \left ({s}\right)$ to $s$

and:
 * '''$f$ carries $s$ into $f \left ({s}\right)$

can be found.

The modifier by $f$ can also be used for under $f$.

Thus, for example, the image of $s$ by $f$ means the same as the image of $s$ under $f$.

In the context of computability theory, the following terms are frequently found:

If $\left({x, y}\right) \in f$, then $y$ is often called the output of $f$ for input $x$, or simply, the output of $f$ at $x$.

Also see

 * Definition:Image of Element under Relation