# Reflexivity, Transitivity, Symmertry of the square of an relation

$\def\p{\mathrel p}$If $\p$ is a relation on a set $A$, define $\p^2$ by $a \mathrel{\p^2} b$ if and only if there exists $c$ with $a \p c$ and $c \p b$.

If $p$ is reflexive/symmetric/transitive does $p^2$ have the same properties?

I’m not even sure how to start this, I assume I would need to use the $a$ related to $c$, $c$ related to $b$ somehow?

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Let’s do reflexivity: Suppose, $p$ is reflexive. Let $a \in A$. We want to show that $a \mathrel{p^2} a$. By the definition of $p^2$, we have to find a $c \in A$ with $a \mathrel p c$, $c \mathrel p a$. But by the reflexivity of $p$, we know that $a \mathrel p a$. So if we let $c = a$, we have $a \mathrel p c$, $c \mathrel p a$, so $a \mathrel{p^2} a$ holds and $p$ is reflexive.