Definition:Relation/Graph

This page is about the graph of a relation. For other uses, see graph.

Definition

Let $S \times T$ be the cartesian product of two sets $S$ and $T$.

Let $\RR$ be a relation on $S \times T$.

The graph of $\RR$ is the set of all ordered pairs $\tuple {s, t}$ of $S \times T$ such that $s \mathrel \RR t$:

$\map \TT \RR = \set {\tuple {s, t}: s \mathrel \RR t}$

Also known as

The graph of a relation is sometimes seen referred to as its truth set.

Also see

• Results about graphs of relations can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 10$: Equivalence Relations
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings
• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.3$: Relations: Example $2.3.1$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): graph (of a relation)