# Antisymmetric Relation

## Antisymmetric Relation

Relation in a set E so that for all ordered pairs (x, y) of E where xy, the ordered pair (y, x) does not belong to E.

In the arrow representation of an antisymmetric relation, if there is one arrow going between two elements, there is no return arrow.

More formally, a relationship ℜ is called antisymmetric when it verifies the following condition: (xy  ∧ yx) x = y.  In other words, if, in a relationship ℜ we have both the ordered pair (x, y) and its inverse pair (y, x), then x and y correspond to the same element.

### Examples

• The relation “…is a proper divisor of…” in the set of whole numbers is an antisymmetric relation.
If 5 is a proper divisor of 15, then 15 cannot be a proper divisor of 5.
• The relation “…has a son…” in a set of people is an antisymmetric relation.
If Paul is Luke’s son, then Luke cannot be Paul’s son.