Difference between revisions of "Binary relation"
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Thus, the relation <math>\sim</math> of [[triangle]] [[similarity]] is a binary relation over the [[set]] of triangles but the relation <math>R(x, y, z) = \{(x, y, z) \mid x, y, z \in \mathbb{Z}_{>0}, x\cdot y = z\}</math> which says <math>x\cdot y</math> is a [[divisor | factor]]ization of <math>z</math> over the [[positive integer]]s is not a binary relation because it takes 3 arguments. | Thus, the relation <math>\sim</math> of [[triangle]] [[similarity]] is a binary relation over the [[set]] of triangles but the relation <math>R(x, y, z) = \{(x, y, z) \mid x, y, z \in \mathbb{Z}_{>0}, x\cdot y = z\}</math> which says <math>x\cdot y</math> is a [[divisor | factor]]ization of <math>z</math> over the [[positive integer]]s is not a binary relation because it takes 3 arguments. | ||
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| + | == Formal Definition and Notation == | ||
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| + | Formally, we say that a relation <math>\mathfrak{R}</math> on sets <math>A</math> and <math>B</math> is a subset of <math>A \times B</math> (the [[Cartesian product]] of <math>A</math> and <math>B</math>). We often write <math>a \, \mathfrak{R} \, b</math> instead of <math>(a,b) \in \mathfrak{R}</math>. | ||
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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]]. | ||
==See also== | ==See also== | ||
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* [[Equivalence relation]] | * [[Equivalence relation]] | ||
| + | * [[Order relation]] | ||
| + | * [[Function]] | ||
* [[Reflexive]] | * [[Reflexive]] | ||
{{stub}} | {{stub}} | ||
Revision as of 12:02, 25 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 
.
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.
See also
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