# Relation

A subset of a finite Cartesian power of a given set , i.e. a set of tuples of elements of .

A subset is called an -place, or an -ary, relation on . The number is called the rank, or type, of the relation . A subset is also called an -place, or -ary, predicate on . The notation signifies that .

One-place relations are called properties. Two-place relations are called binary, three-place relations are called ternary, etc.

The set and the empty subset in are called, respectively, the universal relation and the zero relation of rank on . The diagonal of the set , i.e. the set

is called the equality relation on .

If and are -place relations on , then the following subsets of will also be -place relations on :

The set of all -ary relations on is a Boolean algebra relative to the operations . An -place relation on is called a function if for any elements , from it follows from and that .

See also Binary relation; Correspondence.

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#### References

[a1] | J.L. Bell, M. Machover, "A course in mathematical logic" , North-Holland (1977) |

**How to Cite This Entry:**

Relation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Relation&oldid=14950