ILL 1987 Problem 3

A town has a road network that consists entirely of one-way streets that are used for bus routes. Along these routes, bus stops have been set up. If the one-way signs permit travel from bus stop $X$ to bus stop $Y \neq X$ , then we shall say $Y$ can be reached from $X$. We shall use the phrase $Y$ comes after $X$ when we wish to express that every bus stop from which the bus stop $X$ can be reached is a bus stop from which the bus stop $Y$ can be reached, and every bus stop that can be reached from $Y$ can also be reached from $X$ . A visitor to this town discovers that if $X$ and $Y$ are any two different bus stops, then the two sentences " $Y$ can be reached from $X$ " and " $Y$ comes after $X$ " have exactly the same meaning in this town. Let $A$ and $B$ be two bus stops. Show that of the following two statements, exactly one is true:(i) $B$ can be reached from $A;$ (ii) $A$ can be reached from $B.$