In mathematics, a binary relation

*R* on a set

*X* is

**anti-symmetric** if there is no pair of distinct elements of

*X* each of which is related by

*R* to the other. More formally,

*R* is anti-symmetric precisely if for all

*a* and

*b* in

*X*
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In mathematics, a binary relation *R* on a set *X* is **anti-symmetric** if there is no pair of distinct elements of *X* each of which is related by *R* to the other. More formally, *R* is anti-symmetric precisely if for all *a* and *b* in *X*

or, equivalently,

As a simple example, the divisibility order on the natural numbers is an anti-symmetric relation. And what anti-symmetry means here is that the only way each of two numbers can be divisible by the other is if the two are, in fact, the same number; equivalently, if *n* and *m* are distinct and *n* is a factor of *m*, then *m* cannot be a factor of *n*.

In mathematical notation, this is:

or, equivalently,

The usual order relation ≤ on the real numbers is anti-symmetric: if for two real numbers *x* and *y* both inequalities *x* ≤ *y* and *y* ≤ *x* hold then *x* and *y* must be equal. Similarly, the subset order ⊆ on the subsets of any given set is anti-symmetric: given two sets *A* and *B*, if every element in *A* also is in *B* and every element in *B* is also in *A*, then *A* and *B* must contain all the same elements and therefore be equal:

Partial and total orders are anti-symmetric by definition. A relation can be both symmetric and anti-symmetric (e.g., the equality relation), and there are relations which are neither symmetric nor anti-symmetric (e.g., the "preys on" relation on biological species).

Anti-symmetry is different from asymmetry, which requires both anti-symmetry and irreflexivity.

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