Definition:Relation/General Definition

Definition

Let $\ds \Bbb S = \prod_{i \mathop = 1}^n S_i = S_1 \times S_2 \times \ldots \times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \ldots, S_n$.

An $n$-ary relation on $\Bbb S$ is an ordered $n + 1$-tuple $\RR$ defined as:

$\RR := \struct {S_1, S_2, \ldots, S_n, R}$

where $R$ is an arbitrary subset $R \subseteq \Bbb S$.

To indicate that $\tuple {s_1, s_2, \ldots, s_n} \in R$, we write:

$\map \RR {s_1, s_2, \ldots, s_n}$

Also see

• Results about relations can be found here.

Sources

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 3$: Equivalence relations and quotient sets: Binary relations
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): relation: 3.
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): $n$-ary relation
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): relation: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): $n$-ary relation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): relation: 1.
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.4$: Definition $\text{A}.19$