In mathematics, an asymmetric relation is a binary relation R on a set X where for all a, b \in X, if a is related to b then b is not related to a..
Formal definition
A binary relation on X is any subset R of X \times X. Given a, b \in X, write a R b if and only if (a, b) \in R, which means that a R b is shorthand for (a, b) \in R.
The expression a R b is read as "a is related to b by R."
The binary relation R is called  if for all a, b \in X, if a R b is true then b R a is false; that is, if (a, b) \in R then (b, a) \not\in R.
This can be written in the notation of first-order logic as \forall a, b \in X: a R b \implies \lnot(b R a).
A logically equivalent definition is:
for all a, b \in X, at least one of a R b and b R a is ,
which in first-order logic can be written as: \forall a, b \in X: \lnot(a R b \wedge b R a).
An example of an asymmetric relation is the "less than" relation \,<\, between real numbers: if x < y then necessarily y is not less than x.
The "less than or equal" relation \,\leq, on the other hand, is not asymmetric, because reversing for example, x \leq x produces x \leq x 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.
Properties
A relation is asymmetric if and only if it is both antisymmetric and irreflexive..
Restrictions and converses of asymmetric relations are also asymmetric.
For example, the restriction of \,<\, from the reals to the integers is still asymmetric, and the inverse \,>\, of \,<\, is also asymmetric.
A transitive relation is asymmetric if and only if it is irreflexive: Lemma 1.1 (iv).
Note that this source refers to asymmetric relations as "strictly antisymmetric".
if aRb and bRa, transitivity gives aRa, contradicting irreflexivity.
As a consequence, a relation is transitive and asymmetric if and only if it is a strict partial order.
Not all asymmetric relations are strict partial orders.
An example of an asymmetric non-transitive, even antitransitive relation is the  relation: if X beats Y, then Y does not beat X; and if X beats Y and Y beats Z, then X does not beat Z.
An asymmetric relation need not have the connex property.
For example, the strict subset relation \,\subsetneq\, is asymmetric, and neither of the sets \{1, 2\} and \{3, 4\}is a strict subset of the other.
A relation is connex if and only if its complement is asymmetric.
See also
Tarski's axiomatization of the reals – part of this is the requirement that \,<\, over the real numbers be asymmetric.
References
