|Transitive binary relations
| 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 in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. |
All definitions tacitly require the homogeneous relation be transitive: for all if and then
A term's definition may require additional properties that are not listed in this table.
In mathematics, an asymmetric relation is a binary relation on a set where for all if is related to then is not related to 
A binary relation on is any subset of Given write if and only if which means that is shorthand for The expression is read as " is related to by " The binary relation is called asymmetric if for all if is true then is false; that is, if then
This can be written in the notation of first-order logic as
A logically equivalent definition is:
- for all at least one of and is false,
which in first-order logic can be written as:
An example of an asymmetric relation is the "less than" relation between real numbers: if then necessarily is not less than The "less than or equal" relation on the other hand, is not asymmetric, because reversing for example, produces and both are true.
Asymmetry is not the same thing as "not symmetric": the less-than-or-equal relation is an example of a relation that is neither symmetric nor asymmetric. The empty relation is the only relation that is (vacuously) both symmetric and asymmetric.