Definition:Image (Relation 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:
- $\Img s = \map f s = \ds \bigcup \set {t \in T: \tuple {s, t} \in f}$
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 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.
In the context of set theory, the term image set of mapping for $\Img f$ can often be seen.
Also see
- Definition:Preimage of Mapping (also known as Definition:Inverse Image)
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): image