Definition:Null Relation

Definition

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

$\RR \subseteq S \times T: \RR = \O$

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

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

Also known as

The null relation 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.

Some sources prefer to call it the empty relation.

Also see

• Definition:Trivial Relation

• Results about the null relation can be found here.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: 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 $\text{A}.2$: $11$