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Asymmetric relation
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== Examples == An example of an asymmetric relation is the "[[less than]]" relation <math>\,<\,</math> between [[real number]]s: if <math>x < y</math> then necessarily <math>y</math> is not less than <math>x.</math> More generally, any strict partial order is an asymmetric relation. Not all asymmetric relations are strict partial orders. An example of an asymmetric non-transitive, even [[antitransitive]] relation is the {{em|[[rock paper scissors]]}} relation: if <math>X</math> beats <math>Y,</math> then <math>Y</math> does not beat <math>X;</math> and if <math>X</math> beats <math>Y</math> and <math>Y</math> beats <math>Z,</math> then <math>X</math> does not beat <math>Z.</math> [[Binary relation#Restriction|Restrictions]] and [[Converse relation|converses]] of asymmetric relations are also asymmetric. For example, the restriction of <math>\,<\,</math> from the reals to the integers is still asymmetric, and the converse or dual <math>\,>\,</math> of <math>\,<\,</math> is also asymmetric. An asymmetric relation need not have the [[Connex relation|connex property]]. For example, the [[strict subset]] relation <math>\,\subsetneq\,</math> is asymmetric, and neither of the sets <math>\{1, 2\}</math> and <math>\{3, 4\}</math> is a strict subset of the other. A relation is connex if and only if its complement is asymmetric. A non-example is the "less than or equal" relation <math>\leq</math>. This is not asymmetric, because reversing for example, <math>x \leq x</math> produces <math>x \leq x</math> and both are true. The less-than-or-equal relation is an example of a relation that is neither symmetric nor asymmetric, showing that asymmetry is not the same thing as "not [[Symmetric relation|symmetric]]". The [[empty relation]] is the only relation that is ([[Vacuous truth|vacuously]]) both symmetric and asymmetric.
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