# 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**