# Definition:Image (Set Theory)

## Contents

## 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 operator, 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 $\mathcal R$ is the set:

- $\Img {\mathcal R} := \mathcal R \left [{S}\right] = \set {t \in T: \exists s \in S: \tuple {s, t} \in \mathcal R}$

### Image of an Element

Let $s \in S$.

The **image of $s$ by** (or **under**) **$\mathcal R$** is defined as:

- $\mathcal R \paren s := \set {t \in T: \tuple {s, t} \in \mathcal R}$

That is, $\mathcal R \paren s$ is the set of all elements of the codomain of $\mathcal R$ related to $s$ by $\mathcal R$.

### Image of a Subset

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

Then the **image set (of $X$ by $\mathcal R$)** is defined as:

- $\mathcal R \left [{X}\right] := \set {t \in T: \exists s \in X: \left({s, t}\right) \in \mathcal R}$

## 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 $S$.

Then the **image of $X$ (under $f$)** is defined as:

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

## 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 $\mathcal R$ (or mapping $f$) directly to a subset $A$, those sources prefer to define the mapping induced by $\mathcal R$ or $f$ as a separate concept in its own right.

## Also see

- Definition:Mapping, in which the context of an
**image**is usually encountered.

- 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`