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
:
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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.
Comments
References
[a1] | J.L. Bell, M. Machover, "A course in mathematical logic" , North-Holland (1977) |
Relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relation&oldid=37286