What is a Symmetric Relation?

A relation R on a set A is called symmetric relation if and only if 

∀ a, b ∈ A, if (a, b) ∈ R then (b, a) ∈ R and vice versa i.e., 
∀ a, b ∈ A, (a, b) ∈ R (b, a) ∈ R, 
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 also be present in relation R. If any such bRa is not present for any aRb in R then R is not a symmetric relation.

Example:

Consider set A = {a, b}

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

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“.