Difference between revisions of "Binary relation"
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Thus, in the example of <math>\sim</math> above, we may let <math>\sim</math> be the set of ordered pairs of triangles in the Euclidean plane which are similar to each other. We could also define a relation <math>\leq</math> on the [[power set]] of a set <math>S</math>, so that <math>(A,B) \in \leq</math>, or <math>A\leq B</math>, if and only if <math>A</math> and <math>B</math> are [[subset]]s of <math>S</math> and <math>A</math> is a subset of <math>B</math>. This is a common example of an [[order relation]]. | Thus, in the example of <math>\sim</math> above, we may let <math>\sim</math> be the set of ordered pairs of triangles in the Euclidean plane which are similar to each other. We could also define a relation <math>\leq</math> on the [[power set]] of a set <math>S</math>, so that <math>(A,B) \in \leq</math>, or <math>A\leq B</math>, if and only if <math>A</math> and <math>B</math> are [[subset]]s of <math>S</math> and <math>A</math> is a subset of <math>B</math>. This is a common example of an [[order relation]]. | ||
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+ | == Domain and Range == | ||
+ | |||
+ | The domain of a binary relation <math>\mathfrak{R}</math> over <math>A</math> and <math>B</math>, written <math>\text{Dom}(\mathfrak{R})</math>, is defined to be the set <math>\{x \in A | (\exists y \in B)(x,y) \in \mathfrak{R}\}</math>. It is thus the set of the first components of the ordered pairs in <math>\mathfrak{R}</math>. | ||
+ | |||
+ | The range of a binary relation <math>\mathfrak{R}</math> over <math>A</math> and <math>B</math>, written <math>\text{Ran}(\mathfrak{R})</math>, is defined to be the set <math>\{y \in B | (\exists x \in A)(x,y) \in \mathfrak{R}\}</math>. It is thus the set of the second components of the ordered pairs in <math>\mathfrak{R}</math>. | ||
==See also== | ==See also== |
Revision as of 16:58, 30 November 2007
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.
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.
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
.
See also
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