nLab reflexive symmetric relation

Definition

An reflexive symmetric relation is a binary relation that is reflexive and symmetric.

Examples

• Every equivalence relation is an reflexive symmetric relation which is also a transitive relation.

• The negation of a irreflexive symmetric relation is a reflexive symmetric relation.

• The double negation of every equivalence relation is an irreflexive symmetric relation. In constructive mathematics, this is not guaranteed to be transitive.

In constructive mathematics

In constructive mathematics, the double negation of equality cannot in general be proven to be an equivalence relation, because the denial inequality cannot be proved to be an apartness relation. Instead, the double negation of equality is only a stable reflexive symmetric relation. Indeed, since excluded middle cannot be proven, we could distinguish between the following reflexive symmetric relations in constructive mathematics:

• equality

• equivalence relations, a transitive reflexive symmetric relation

• double negation of equality

• stable reflexive symmetric relation, a reflexive symmetric relation $\approx$ which additionally satisfies $\neg \neg (a \approx b)$ implies $a \approx b$

• decidable reflexive symmetric relation, an reflexive symmetric relation $\approx$ which additionally satisfies $(a \approx b)$ or $\neg (a \approx b)$

 See also

• double negation
• irreflexive symmetric relation
• equivalence relation