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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+ | More generally, we say that a relation <math>\mathfrak{R}(x,y)</math> is a mathematical sentence in which two letters, <math>x</math> and <math>y</math>, are of particular interest. This more general definition is useful because it admits relations whose "domain" is a class of sets too large to constitute a set. For instance, the relation <math>\mathfrak{R}(x,y)</math> defined as <math>(x=y)</math> applies to all sets, not just sets contained in some larger set. | ||
== Domain and Range == | == Domain and Range == | ||
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A binary relation <math>\mathfrak{R}</math> over <math>A</math> is defined to be transitive if <math>(\forall (a,b,c) \in A^3)(((a \mathfrak{R} b) \wedge (b \mathfrak{R} c)) \rightarrow (a \mathfrak{R} c))</math>. | A binary relation <math>\mathfrak{R}</math> over <math>A</math> is defined to be transitive if <math>(\forall (a,b,c) \in A^3)(((a \mathfrak{R} b) \wedge (b \mathfrak{R} c)) \rightarrow (a \mathfrak{R} c))</math>. | ||
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+ | 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== | ==See also== |
Latest revision as of 22:33, 18 May 2008
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. This more general definition is useful because it admits relations whose "domain" is a class of sets too large to constitute a set. For instance, the relation
defined as
applies to all sets, not just sets contained in some larger set.
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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