Definition:Image (Set Theory)/Mapping/Element

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Definition

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

Let $s \in S$.

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

$\Img s = \map f s = \displaystyle \bigcup \set {t \in T: \tuple {s, t} \in f}$

That is, $\map f s$ is the element of the codomain of $f$ related to $s$ by $f$.


By the nature of a mapping, $f \paren s$ is guaranteed to exist and to be unique for any given $s$ in the domain of $f$.


Also known as

The image of an element $s$ under a mapping $f$ is also called the functional value, or value, of $f$ at $s$.

The terminology:

$f$ maps $s$ to $\map f s$
$f$ assigns the value $\map f s$ to $s$
$f$ carries $s$ into $\map f s$

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 $\tuple {x, y} \in f$, then $y$ is often called the output of $f$ for input $x$, or simply, the output of $f$ at $x$.


Also denoted as

The notation $\Img f$ to denote the image of an object $f$ is specific to $\mathsf{Pr} \infty \mathsf{fWiki}$.

The usual notation is $\image f$ or a variant, but this is too easily confused with $\map \Im z$, the imaginary part of a complex number.

Hence the non-standard usage $\Img f$.


The notation $\Img f$ is specific to $\mathsf{Pr} \infty \mathsf{fWiki}$.


Examples

Image of $2$ under $x \mapsto x^4 - 1$

Let $f: \R \to \R$ be the mapping defined as:

$\forall x \in \R: f \paren x = x^4 - 1$

The image of $2$ is:

$f \paren 2 = 15$


Also see


Sources