- May 2020
R is antisymmetric iff ∀∀\forall x ∀∀\forall y [xRy ∧ yRx → x = y].
For any x and y in a set, if x is related to y and y is related to x, for it to be antisymmetric x must equal y.
We have the set [1 2 3].
For a relation to be antisymmetric, if we have (1 2) we must not have (2 1). In other words, if there is any symmetry it must be between the same elements, such as (1 1).
Additionally, if we do not have any element related to any element it is vacuously antisymmetric.