# Antitransitive Relation

## Antitransitive Relation

Relation in a set E so that for the ordered pairs (x, y) and (y, z) to belong to the graph, the ordered pair (x, z) must not.

In the arrow representation of an antitransitive relation, every time that we have the ordered pairs (x, y) and (y, z), we do not have the ordered pair (x, z).

In this representation, for example, we have: (a, b) and (b, c), but not (a, c).

### Examples

• The relation “…is perpendicular to…” in a set of lines of the plane is an antitransitive relation.
If d ⊥ d$$_{1}$$ and d$$_{1}$$⊥ d$$_{2}$$, it is impossible that d ⊥ d$$_{2}$$, because d  ⁄ ⁄  d$$_{2}$$.
• The relation “…is the mother of…” in a set of people is an antitransitive relation.
If Anne is Julie’s mother and Julie is Marie’s mother, then Anne is Marie’s grandmother and cannot be Marie’s mother.