Definition:Image (Relation Theory)
Definition
The definition of a relation as a subset of the Cartesian product of two sets gives a "static" sort of feel to the concept.
However, we can also consider a relation as being an operation, where you feed an element $s \in S$ (or a subset $S_1 \subseteq S$) in at one end, and you get a set of elements $T_s \subseteq T$ out of the other.
Thus we arrive at the following definition.
Relation
Image of a Relation
The image of $\RR$ is defined as:
- $\Img \RR := \RR \sqbrk S = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}$
Image of an Element
Let $s \in S$.
The image of $s$ by (or under) $\RR$ is defined as:
- $\map \RR s := \set {t \in T: \tuple {s, t} \in \RR}$
That is, $\map \RR s$ is the set of all elements of the codomain of $\RR$ related to $s$ by $\RR$.
Image of a Subset
Let $X \subseteq S$ be a subset of $S$.
Then the image set (of $X$ by $\RR$) is defined as:
- $\RR \sqbrk X := \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR}$
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:
- $\Img s = \map f s = \ds \bigcup \set {t \in T: \tuple {s, t} \in f}$
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 (usually) mapping, in order to differentiate it from an inverse image.
Rather than apply a relation $\RR$ (or mapping $f$) directly to a subset $A$, those sources prefer to define the direct image mapping of $f$ as a separate concept in its own right.
Also see
- Definition:Mapping, in which the context of an image is usually encountered.
- Definition:Domain (Relation Theory)
- Definition:Codomain (Relation Theory)
- Definition:Range of Relation
- Definition:Preimage (also known as Definition:Inverse Image)
Technical Note
The $\LaTeX$ code for \(\Img {f}\) is \Img {f} .
When the argument is a single character, it is usual to omit the braces:
\Img f