I need a relation which is not reflexive, not symmetric, and not transitive

Solution 1:

Here's a non-mathematical one: "is the father of".
You are not your own father. You are not your father's father. Your father's father is not your father.

Solution 2:

Think of three points $u, v, w$ with relation $R = \{(u, v), (v, w) \}$. So $u$ is related to $v$ and $v$ is related to $w$. This is not reflexive since $(u,u) \notin R$, not symmetric because $(v, u) \notin R$ and not transitive because $(u, w) \notin R$.

Solution 3:

What beats what in Roshambo or "Rock, Paper, Scissors" is such a relation.

1. not reflexive: rock does not beat rock.

2. not symmetric: rock beats scissors, but scissors does not beat rock.

3. 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".