Definition:Domain (Relation Theory)/Relation
Definition
Let $\RR \subseteq S \times T$ be a relation.
The domain of $\RR$ is defined and denoted as:
- $\Dom \RR := \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$
That is, it is the same as what is defined here as the preimage of $\RR$.
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 domain of $\RR$ is the set defined as:
- $\ds \Dom \RR := \set {\tuple {s_1, s_2, \ldots, s_{n - 1} } \in \prod_{i \mathop = 1}^{n - 1} S_i: \exists s_n \in S_n: \tuple {s_1, s_2, \ldots, s_n} \in \RR}$
The concept is usually encountered when $\RR$ is an endorelation on $S$:
- $\ds \Dom \RR := \set {\tuple {s_1, s_2, \ldots, s_{n - 1} } \in S^{n - 1}: \exists s_n \in S_n: \tuple {s_1, s_2, \ldots, s_n} \in \RR}$
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 domain of $\RR$ is defined and denoted as:
- $\Dom \RR := \set {x \in V: \exists y \in V: \tuple {x, y} \in \RR}$
That is, it is the class of all $x$ such that $\tuple {x, y} \in \RR$ for at least one $y$.
Also defined as
It is usual, as here, to define the domain of $\RR \subseteq S \times T$ as the subset of $S$ that bears some element of $S$ to $T$.
However, it appears to make sense to define it to be the whole of the set $S$.
Using this definition, $s \in \Dom \RR$ whether or not $\exists t \in T: \tuple {s, t} \in \RR$.
It would then be possible to refer to $\set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$ as the preimage of $\RR$.
This appears never to be done in the literature that has so far been studied as source work for $\mathsf{Pr} \infty \mathsf{fWiki}$.
Most texts do not define the domain in the context of a relation in the first place, so this question does not often arise.
Even if it does, the domain is often such that either it coincides with $S$ or that it appears to be of small importance.
Also known as
Some sources refer to the domain of $\RR$ as the domain of definition of $\RR$.
Some sources use a distinctive typeface, for example, $\map {\mathsf {Dom} } \RR$.
Also see
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of 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: $\S 6.5$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings
- 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: $1.5$ Relations
- 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 $\text{A}.23$