# Definition:Null Relation

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## Contents

## Definition

The **null relation ** is a relation $\mathcal R$ in $S$ to $T$ such that $\mathcal R$ is the empty set:

- $\mathcal R \subseteq S \times T: \mathcal R = \O$

That is, *no* element of $S$ relates to *any* element in $T$:

- $\mathcal R: S \times T: \forall \tuple {s, t} \in S \times T: \neg s \mathrel {\mathcal R} t$

## Also known as

This is also sometimes referred to as **a trivial relation** by some authors, but to save confusion it is better to use that term specifically to mean this one.

Other sources prefer to call it the **empty relation**.

## Also see

- Results about
**the null relation**can be found here.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 7$: Relations - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 4$. Relations; functional relations; mappings: Example $4.3$ - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations: Problem Set $\text{A}.2$: $11$