Anti-symmetric Relation on Set
What is anti-symmetric relation onset?
Let A be a set in which the relation R defined.
R is said to be anti-symmetric,
if there exist elements, if
aRb and
bRa ⇒ a = b
that is,
(a, b) ∈ R
and ((b, a) ∈ R ⇒ a = b.
A relation
R in A is not anti-symmetric,
if there exist elements a, b ∈ A, a ≠ b such that aRb and bRa.
For
example, the relation defined by ‘x is less than or equal to’ in the set of
real numbers is be anti-symmetric,
as a ≤ b and b ≤ a imply a = b,
where a, b are elements of the set.