For vectors $(x_1,x_2,\dots,x_n), (y_1,y_2,y_n)$, is "$\forall k$ s.t. $x_k>y_k$, $x_k-y_k>C$ vacuously true when $x_k\leq y_k \ \forall k$? Where C is some positive constant?
Basically, I am confused regarding vacuously true statements and trying to use this constructed example to clarify.
The restate the example in slightly more detail, suppose we have two vectors denoted $(x_1,x_2,\dots,x_n), (y_1,y_2,y_n)$.
The statement in consideration is that, "$\forall k$ , if $x_k>y_k$, then $x_k-y_k>C$.
If it is the case that $x_k\leq y_k \ \  \forall k$, then the statement is vacuously true, correct?


*

*Basically, is any statement automatically true on an empty set 

*(to continue with the above example, if I say "for all vector indices on which $x$ exceeds $y$, the amount by which $x$ exceeds $y$ on that index is at least C), this statement doesn't guarantee that any such indices exist, so for the statement to be meaningful I would need to address this)

 A: Vacuous truth
The phrase vacuous truth is informally used to describe statements of the form (1), which are true because the set $X$ is empty. 
\begin{equation}\tag{1}
\forall x\in X.P(x)
\end{equation}
It can also refer to statements of the form (2) that happen to be true because no $x\in X$ satisfies Q(x). (See principal of explosion).
\begin{equation}\tag{2}
\forall x\in X.Q(x)\rightarrow P(x)
\end{equation}
Principal of explosion
\begin{equation}\tag{3}
A\rightarrow B
\end{equation}
Statements of the form (2), are always true whenever $A$ is false. This is due to the definition of implication in symbolic logic.
\begin{array}
{cccc}&A&\lnot A\\
~B &\color{blue}{1}&\color{blue}{1}\\
\lnot B~&\color{red}{0}&\color{blue}{1}\\
\end{array}
The above truth table of (3) shows when an implication is true or false. The justification is that (3) should only be false when its premises is true, and its consequent is false. 
The principal of explosion states that any statement can be proven from a contradiction (falsehood). If $A$ is false in (3), then we have assumed falsehood, and anything may be derived including $B$.
Your example
\begin{equation}\tag{4}
\forall k\in K. (x_k>y_k)\rightarrow (x_k−y_k>C)
\end{equation}
Often the principal of explosion and vacuous truth are present together. For instance, if the set $X$ is empty in (2), then $\forall x\in X.\lnot Q(x)$ is true vacuously (no x in X satisfies Q(x)), and (2) is then true by the principal of explosion. 
If $x_k\leq y_k$, then (4) is true by the principal of explosion. It is also vacuously true, since it can be proven that no $k$ satisfy the premises of (4).
