Binary relation
A binary relation is a relation which relates pairs of objects.
Thus, the relation of triangle similarity is a binary relation over the set of triangles but the relation
which says
is a factorization of
over the positive integers is not a binary relation because it takes 3 arguments.
Contents
Formal Definition and Notation
Formally, we say that a relation on sets
and
is a subset of
(the Cartesian product of
and
). We often write
instead of
. If
(the case of most common interest), then we say that
is a relation on
.
Thus, in the example of above, we may let
be the set of ordered pairs of triangles in the Euclidean plane which are similar to each other. We could also define a relation
on the power set of a set
, so that
, or
, if and only if
and
are subsets of
and
is a subset of
. This is a common example of an order relation.
More generally, we say that a relation is a mathematical sentence in which two letters,
and
, are of particular interest.
Domain and Range
The domain of a binary relation over
and
, written
, is defined to be the set
. It is thus the set of the first components of the ordered pairs in
.
The range of a binary relation over
and
, written
, is defined to be the set
. It is thus the set of the second components of the ordered pairs in
.
Reflexivity, Symmetry and Transitivity
A binary relation over
is defined to be reflexive if
.
A binary relation over
is defined to be symmetric if
.
A binary relation over
is defined to be anti-symmetric if
.
A binary relation over
is defined to be transitive if
.
A reflexive, symmetric and transitive relation is called an equivalence relation. A reflexive, anti-symmetric and transitive relation is called an order relation.
See also
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