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