What is an Anti-Symmetric Relation?

A relation R on a set A is called anti-symmetric relation if 

∀ a, b ∈ A, if (a, b) ∈ R then (b, a) ∉ R or a = b, 
where R is a subset of (A x A), i.e. the cartesian product of set A with itself.

This means if an ordered pair of elements “a” to “b” (aRb) is present in relation R then an ordered pair of elements “b” to “a” (bRa) should not be present in relation R unless a = b.

If any such bRa is present for any aRb in R then R is not an anti-symmetric relation.

Example:

Consider set A = {a, b}

R = {(a, b), (b, a)} is not anti-symmetric relation as for (a, b) tuple (b, a) tuple is present but
R = {(a, a), (a, b)} is an anti-symmetric relation.

A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on. To learn more about relations refer to the article on “Relation and their types“.