TITLE: I need a relation which is not reflexive, not symmetric, and not transitive
QUESTION [13 upvotes]: I need an example of a relation which is simultaneously not reflexive, not symmetric, and not transitive.  Any accessible examples?  Thanks in advance.

REPLY [3 votes]: Take any directed acyclic graph amd the arcs form an irreflexive, asymmetric antitransitive relation of its nodes. Then add some loops (not to all nodes), back-arcs (not to all of them) and some skip-forward arcs (not to all directed paths) and you have a more general relation with your restrictions.
Ex: 
1) Strong version: a->b, b->c, c->d, a->e
2) Then add: a->a, b->a,a->d