Definition:Image (Set Theory)/Mapping

Definition

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

Image of a Mapping

Definition 1

The image of a mapping $f: S \to T$ is the set:

$\Img f = \set {t \in T: \exists s \in S: \map f s = t}$

That is, it is the set of values taken by $f$.

Definition 2

The image of a mapping $f: S \to T$ is the set:

$\Img f = f \sqbrk S$

where $f \sqbrk S$ is the image of $S$ under $f$.

Image of an Element

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 $f$.

Image of a Subset

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

Let $X \subseteq S$ be a subset of $S$.

The image of $X$ (under $f$) is defined and denoted as:

$f \sqbrk X := \set {t \in T: \exists s \in X: \map f s = t}$

Also known as

Some sources refer to this as the direct image of a mapping, in order to differentiate it from an inverse image.

Rather than apply a mapping $f$ directly to a subset $A$, those sources often prefer to define the direct image mapping of $f$ as a separate concept in its own right.