# Definition:Image (Set Theory)/Relation/Relation

## Definition

Let $\mathcal R \subseteq S \times T$ be a relation.

The image of $\mathcal R$ is the set:

$\Img {\mathcal R} := \mathcal R \sqbrk S = \set {t \in T: \exists s \in S: \tuple {s, t} \in \mathcal R}$

### General Definition

Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$.

Let $\displaystyle \mathcal R \subseteq \prod_{i \mathop = 1}^n S_i$ be an $n$-ary relation on $\displaystyle \prod_{i \mathop = 1}^n S_i$.

The image of $\mathcal R$ is the set defined as:

$\Img {\mathcal R} := \set {s_n \in S_n: \exists \tuple {s_1, s_2, \ldots, s_{n - 1} } \in \displaystyle \prod_{i \mathop = 1}^{n - 1} S_i: \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$

The concept is usually encountered when $\mathcal R$ is an endorelation on $S$:

$\Img {\mathcal R} := \set {s_n \in S: \exists \tuple {s_1, s_2, \ldots, s_{n - 1} } \in S^{n - 1}: \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$

## Also known as

The image of $\mathcal R$ is often seen referred to as the image set of $\mathcal R$.

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

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

Other sources call the image of $\mathcal R$ its range, but this convention is discouraged because of potential confusion.

Many sources denote the image of a relation $\mathcal R$ by $\map {\operatorname {Im} } {\mathcal R}$, but this notation can be confused with the imaginary part of a complex number $\map \Im z$.

Hence on $\mathsf{Pr} \infty \mathsf{fWiki}$ it is preferred that $\Img {\mathcal R}$ be used.

## 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