Definition:Relation
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Definition
Let $S \times T$ be the cartesian product of two sets $S$ and $T$.
A relation (in this context, technically speaking, a binary relation) on $S \times T$ is an arbitrary subset $\mathcal R \subseteq S \times T$.
What this means is that a binary relation relates (certain) elements of one set with (certain) elements of another.
Not all elements in $S$ need to be related to every relation in $T$ (but see Trivial Relation, which is a relation in which they are).
When $\left({s, t}\right) \in \mathcal R$, we can write:
- $s \mathcal R t$
or:
- $\mathcal R \left({s, t}\right)$.
and can say $s$ bears $\mathcal R$ to $t$.
If $\left({s, t}\right) \notin \mathcal R$, we can write: $s \not \mathcal R t$, that is, by drawing a line through the relation symbol. See Complement of Relation.
Truth Set
Let $\mathcal R \subseteq S \times T$ be a relation.
The truth set of $\mathcal R$ is the set of all ordered pairs of $\mathcal R$:
- $\mathcal T \left({\mathcal R}\right) = \left\{{\left({s, t}\right): s \mathcal R t}\right\}$
Relation as a Mapping
It is possible to define a relation as a mapping from the cartesian product $S \times T$ to a boolean domain $\left\{{\text{true}, \text{false}}\right\}$:
- $\mathcal R: S \times T \to \left\{{\text{true}, \text{false}}\right\}: \forall \left({s, t}\right) \in S \times T: \mathcal R \left({s, t}\right) = \begin{cases} \text{true} & : \left({s, t}\right) \in \mathcal R \\ \text{false} & : \left({s, t}\right) \notin \mathcal R \end{cases}$
but this is too unwieldy and overcomplicated to be practical. It also relies on a circular definition. However, it can have the advantage of making the concept clear.
This approach is made in Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951).
Endorelation
Let $S \times S$ be the cartesian product of a set $S$ with itself.
Let $\mathcal R \subseteq S \times S$ be a relation on $S \times S$
Then $\mathcal R$ is referred to as an endorelation, or a relation in $S$, or a relation on $S$.
Some sources use the term binary relation exclusively to refer to a binary endorelation.
Generalized Definition
Let $\displaystyle \Bbb S = \prod_{i=1}^n S_i = S_1 \times S_2 \times \ldots \times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \ldots, S_n$.
An arbitrary subset $\mathcal R \subseteq \Bbb S$ is a called an $n$-ary relation on $\Bbb S$.
To indicate that $\left({s_1, s_2, \ldots, s_n}\right) \in \mathcal R$, we write $\mathcal R \left({s_1, s_2, \ldots, s_n}\right)$.
Unary Relation
As a special case of an $n$-ary relation on $S$, note that when $n = 1$ we define a unary relation on $S$ as:
- $\mathcal R \subseteq S$
... that is, as a subset of $S$.
Also see
- Results about relations can be found here.
Linguistic Note
In natural language what we have defined as a relation is usually understood as a relationship.
Sources
- Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 3$
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 7$: Relations
- W.E. Deskins: Abstract Algebra (1964): $\S 1.2$: Definition $1.5$
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $2.1$
- Seth Warner: Modern Algebra (1965): $\S 10$
- Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.3$
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{I}$
- A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.4$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 10 \alpha$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4, \S 6$
- Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.2$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 15$
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$
- John F. Humphreys: A Course in Group Theory (1996): $\S 2$: Definition $2.20$
