Definition:Image (Relation Theory)/Relation/Relation

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Let $\RR \subseteq S \times T$ be a relation.

The image of $\RR$ is defined as:

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

General Definition

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

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

The image of $\RR$ is the set defined as:

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

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

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

Class Theory

In the context of class theory, the definition follows the same lines:

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation in $V$.

The image of $\RR$ is defined and denoted as:

$\Img \RR := \set {y \in V: \exists x \in V: \tuple {x, y} \in \RR}$

That is, it is the class of all $y$ such that $\tuple {x, y} \in \RR$ for at least one $x$.

Also known as

The image of a relation $\RR$, when in the context of set theory, is often seen referred to as the image set of $\RR$.

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 $\RR$ directly to a subset $A$, those sources often prefer to define the direct image mapping of $\RR$ as a separate concept in its own right.

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

Many sources denote the image of a relation $\RR$ by $\map {\operatorname {Im} } \RR$, 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 $\Img \RR$ be used.

Also see

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