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 == | ||
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| + | 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>. | ||
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| + | 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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