Relation Glossary
Definition
An n-ary relation is a property of n-tuples of objects. In the semantics of first order logic a relation is identified with its extension—i.e. the set of n-tuples related by it.
By slight abuse of terminology, we may refer to an n-ary predicate as a relation, especially where n > 1. It is common to call a predicate symbol a "relation symbol" in such a case.
Comments
Binary relations are especially important, as they commonly occur in almost all uses of logic. They can be studied as objects in their own right. Some binary relations, such as identity and the ordering relations < and > in Logic for Fun are important enough to be regarded as logical constants with their own introduction and elimination rules and fixed semantics.
The logic of relations is richer than the logic of unary predicates. The ability to deal with relations is an important respect in which modern logic is an advance on earlier systems.
Examples
- The relation of inclusion between sets, written '⊆', is a binary relation which is reflexive, transitive and antisymmetric.
- The relation of non-identity, written '≠', is a binary relation which is irreflexive and symmetric.
- The relation of being between is a ternary relation: e.g. Goulburn is between Canberra and Sydney; 22 is between 20 and 30.