# Definition:Relation/Truth Set

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## 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 **truth set** 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 **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): $\S 4$. 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 $2.3.1$