Antisymmetric relation

In mathematics, a binary relation "R" on a set "X" is antisymmetric if, for all "a" and "b" in "X", if "a" is "R" to "b" and "b" is "R" to "a", then "a" = "b".

In mathematical notation, this is:

:$forall\; a,\; b\; in\; X,\; a\; R\; b\; and\; b\; R\; a\; ;\; Rightarrow\; ;\; a\; =\; b$

or equally,

:$forall\; a,\; b\; in\; X,\; a\; R\; b\; and\; a\; e\; b\; Rightarrow\; lnot\; b\; R\; a.$

Inequalities are antisymmetric, since for numbers "a" and "b", "a ≤ b" and "b ≤ a" if and only if "a = b". The same holds for subsets.

Note that 'antisymmetric' is not the logical negative of 'symmetric' (whereby "aRb" implies "bRa"). (N.B.: Both are properties of relations expressed as universal statements about their members; their logical negations must be existential statements.) Thus, there are relations which are 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 asymmetric, anything that fails to be symmetric is asymmetric. Another definition of asymmetric makes asymmetry equivalent to antisymmetry plus irreflexivity.

Examples

* The equality relation = on any given domain.
* The usual order relation ≤ on the real numbers.
* The subset order ⊆ on the subsets of any given set.
* The relation "x" is even, "y" is odd" between a pair ("x", "y") of integers::::

Properties containing antisymmetry

* Partial order - An antisymmetric relation that is also transitive and reflexive.

* Total order - An antisymmetric relation that is also transitive and total.

ee also

* Symmetry in mathematics
* Symmetric relation
* antisymmetry in linguistics
* nonsymmetric relation
* asymmetric relation