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. | ||
== Domain and Range == | == Domain and Range == |
Revision as of 21:30, 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.
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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