TITLE: Explain phrase "For each pair of distinct elements" in reference to a graph.
QUESTION [6 upvotes]: In one of my algorithm courses, there is this: 

A subset $S$ of vertices in a directed graph $G$ is
  strongly connected if for each pair of distinct
  vertices ($v_i$, $v_j$) in $S$, $v_i$ is connected to $v_j$ and
  $v_j$ is connected to $v_i$.

And then the following example graph is given for this proposition: 

Maybe i do not understand what that phrase means. What i think it means is this: A node, say $E$ can be a member of an ordered pair $(v_i, v_j)$ only once, ie. if $(E, A)$, then $\lnot (E, [someOtherNode]) $. But we clearly see here, that we have $(E,A)$ and $(E,D)$. 
How should i correctly interpret this phrase. What does it mean exactly? Thanks.

REPLY [3 votes]: From your subset $S$ of vertices you can construct a set of pairs $P$, where $(v,w) \in P$ if there is a directed path from $v$ to $w$ (not necessarily confined to $S$).  The set is strongly connected if
$$
    \forall v,w\in S, ( v\neq w \Rightarrow (v,w) \in P \wedge (w,v) \in P)
$$
Let $S=G$ in the example graph.  Notice that $(A,B)$, $(B,C)$, $(C,D)$, $(D,E)$, $(E,A)$ are all in $P$.  So by transitivity, every pair of distinct vertices is in $P$.
The word distinct in the definition rules out the requirement that $(v,v) \in P$.  However, it seems that $(v,w) \in P \wedge (w,v) \in P$ would imply that $(v,v)\in P$ anyway.
I'm not sure what you mean by “valid” pair.  Any pair may be in $P$; the condition is only that certain pairs must be in $P$.