Definition:Image (Relation Theory)/Relation/Relation
Definition
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
- Definition:Mapping, in which the context of an image is usually encountered.
- Definition:Preimage of Relation (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
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.3$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.11$: Relations: Definition $11.1$
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.6$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Relations
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.4$: Definition $\text{A}.23$