Antireflexive Relation/Examples/Strict Ordering

Example of Antireflexive Relation

The relation $<$ on one of the standard number systems $\N$, $\Z$, $\Q$ and $\R$ is antireflexive.

Proof

We have:

$\forall a \in \N: \lnot \paren {a < a}$

Hence the result by definition of antireflexive relation.

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.4$: Equivalence relations: Exercise $1 \ \text{(ii)}$
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Exercise $4 \ \text{(b)}$