I need an example of a relation which is simultaneously not reflexive, not symmetric, and not transitive. Any accessible examples? Thanks in advance.
What beats what in Roshambo or "Rock, Paper, Scissors" is such a relation.
not reflexive: rock does not beat rock.
not symmetric: rock beats scissors, but scissors does not beat rock.
not transitive: rock beats scissors and scissors beats paper, but rock does not beat paper.
The same is true of "Rock, Paper, Scissors, Lizard, Spock".
How about: "is the square of", defined on the set of positive integers? In other words, $$a \sim b \iff a=b^2$$
This relation is not reflexive (most numbers are not their own square), not symmetric (if $a$ is the square of $b$ then in most cases $b$ is not the square of $a$) and not transitive (if $a$ is the square of $b$ and $b$ is the square of $c$ then in general $a$ will not be the square of $c$).
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