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.

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