Let us consider the relation $\sim$ defined upon $ℕ$ as follows: \begin{align} x \sim y \iff x \text{ and } y \text{ are both prime.} \end{align}
What I cannot grasp is the fact that this relation is symmetric and transitive but not reflexive? I asked my lecturer and he gave me the example of reflexivity as in, if we considered $x \sim x$ and took $x$ to be $4,$ we know that $4$ is not prime and, therefore, $4$ is not related to $4.$
So why is it that, if we took any number for $x$ and $y,$ the relation is still symmetric and transitive? I seriously cannot understand this and it's hurting my head.
Any clarification would be sincerely appreciated.
Kindest regards
If $x\sim y$ then it means both $x,y$ are prime. But of course this implies that both $y,x$ are prime (as these are the same numbers, we just wrote them in different order), and so $y\sim x$ by definition. So the relation is symmetric.
As for transitive, suppose $x\sim y$ and $y\sim z$. Then $x,y$ are both prime and $y,z$ are both prime. It follows that $x,z$ are both prime, and so $x\sim z$.