Antisymmetric relation
Binary relation such that if A is related to B and is different from it then B is not related to A / From Wikipedia, the free encyclopedia
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In mathematics, a binary relation on a set is antisymmetric if there is no pair of distinct elements of each of which is related by to the other. More formally, is antisymmetric precisely if for all
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Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation be transitive: for all if and then |
or equivalently,
The definition of antisymmetry says nothing about whether actually holds or not for any . An antisymmetric relation on a set may be reflexive (that is, for all ), irreflexive (that is, for no ), or neither reflexive nor irreflexive. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.