= Solution
Allow a zero-length <graph path>, as in Question 5. Then every <vertex> $a$ reaches itself, proving <reflexivity>. If $a\equiv b$, its definition already includes both directions of <reachability>, so interchanging $a,b$ proves <symmetry>.
If $a\equiv b$ and $b\equiv c$, concatenate the <graph paths> $a\to b$ and $b\to c$ to obtain <reachability> from $a$ to $c$. Similarly concatenate $c\to b$ and $b\to a$ for the reverse direction. Repeated <vertices> can be removed if <graph paths> are required to be simple. Thus $a\equiv c$, proving <transitivity>. Consequently
$$
\boxed{\equiv\text{ is an equivalence relation, whose classes are the strongly connected components.}}
$$
The zero-length convention matters for an isolated <vertex>: it is still a singleton <strongly connected component>.
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