# 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. |