In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X
or, equivalently,
In mathematical notation, this is:
or equally,
An example of an antisymmetric relation is the subset relation:
Or in words, if every element in A also is in B and all elements in B are in A, than A and B must be equal, i.e. containing all the same elements.
Partial and total orders are antisymmetric by definition. Therefore the usual order r...
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In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X
or, equivalently,
In mathematical notation, this is:
or equally,
An example of an antisymmetric relation is the subset relation:
Or in words, if every element in A also is in B and all elements in B are in A, than A and B must be equal, i.e. containing all the same elements.
Partial and total orders are antisymmetric by definition. Therefore the usual order relation ≤ on the real numbers, the subset order ⊆ on the subsets of any given set and the divisibility order of the natural numbers are antisymmetric. For example, if for two real numbers x and y both inequalities x ≤ y and y ≤ x hold then x and y must be equal.
A relation can be both symmetric and antisymmetric (e.g., the equality relation), and there are relations which are neither symmetric nor antisymmetric (e.g., the preys-on relation on biological species).
Antisymmetry is different from asymmetry. According to one definition of...
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