Difference between revisions of "Reflexive property"
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| − | A [[binary relation]] <math>\mathcal R</math> on a [[set]] <math>S</math> is said to be '''reflexive''' if <math>a{\mathcal R}a</math> for all <math>a \in S</math>. | + | A [[binary relation]] <math>\mathcal R</math> on a [[set]] <math>S</math> is said to be '''reflexive''' or to have the '''reflexive property''' if <math>a{\mathcal R}a</math> for all <math>a \in S</math>. |
For example, the relation of [[similarity]] on the set of [[triangle]]s in a [[plane]] is reflexive: every triangle is similar to itself. However, the relation <math>\mathcal R</math> on the [[real number]]s given by <math>x {\mathcal R} y</math> if and only if <math>x < y</math> is not reflexive because <math>x < x</math> does not hold for at least one real value of <math>x</math>. (In fact, it does not hold for any real value of <math>x</math>, but we only need the weaker statement to disprove reflexivity.) | For example, the relation of [[similarity]] on the set of [[triangle]]s in a [[plane]] is reflexive: every triangle is similar to itself. However, the relation <math>\mathcal R</math> on the [[real number]]s given by <math>x {\mathcal R} y</math> if and only if <math>x < y</math> is not reflexive because <math>x < x</math> does not hold for at least one real value of <math>x</math>. (In fact, it does not hold for any real value of <math>x</math>, but we only need the weaker statement to disprove reflexivity.) | ||
Latest revision as of 11:51, 22 May 2007
A binary relation 
on a set 
is said to be reflexive or to have the reflexive property if 
for all 
.
For example, the relation of similarity on the set of triangles in a plane is reflexive: every triangle is similar to itself. However, the relation on the real numbers given by 
if and only if 
is not reflexive because 
does not hold for at least one real value of 
. (In fact, it does not hold for any real value of , but we only need the weaker statement to disprove reflexivity.)
See also
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