# Definition:Relation/Truth Set

< Definition:Relation(Redirected from Definition:Graph of Relation)

## Definition

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

Let $\mathcal R$ be a relation on $S \times T$.

The **truth set** of $\mathcal R$ is the set of all ordered pairs $\tuple {s, t}$ of $S \times T$ such that $s \mathrel {\mathcal R} t$:

- $\map {\mathcal T} {\mathcal R} = \set {\tuple {s, t}: s \mathrel {\mathcal R} t}$

## Also known as

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

However, this term is most usually seen in the context of a mapping.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 10$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 4$ - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.3$: Relations: Example $2.3.1$