# Relations & Concepts

A fact is a statement that is True. Joe Smith lives in New York could be a fact. In this example, Joe Smith could be seen as an instance of a Person and New York as an instance of a City. The following Ampersand terminology applies here:

• Person and City are concepts
• Joe Smith is an atom of the concept Person
• lives in is the relation name of a relation with the sign [Person, City]

## Properties of relations

In many cases, there can be restrictions on the population of a relation, called properties.

The following properties can be specified on any relation r[A*B]

& property semantics
UNI univalent For any a in A there can only be one b in B in the population of r.
INJ injective For any b in B there can only be one a in A in the population of r.
SUR surjective For any b in B there must be (at least) one a in A in the population of 'r.
TOT total For any a in A there must be (at least) one b in B in the population of 'r.

There are additional relations that can be specified on endo relations. An endo relation is a relation where the source and target concepts are equal. r[A*A].

& property semantics
SYM symmetric For each (a,b) in r, (b,a) is in r.
ASY antisymmetric If (a,b) and (b,a) are both in r, then a = b
TRN transitive If (a,b) and (b,c) are both in r, then (a,c) is in r.
RFX reflexive For each a in A, the pair (a,a) is in the population of r
IRF irreflexive For each a in A, the pair (a,a) is not in the population of r
PROP - shortcut for the combination of symmetric and antisymmetric.