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

## Definition

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

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 8$ Relations