How's it possible for each element of the empty set to be even? I was reading Pugh's Real Analysis:
I've found this in the beginning of the book:

A class is a collection of sets. The sets are members of the class. For example we could consider the class $\mathcal{E}$ of sets of even natural numbers. Is the set $\{2,15\}$ a member of $\mathcal{E}$? No. How about the singleton $\{6\}$? Yes. How about the empty set? $\color{red}{\text{Yes, each element of the empty set is even.}}$

How's it possible that each element of the empty set is even when the empty set doesn't have any elements?
Edit: I've read the comments and the answer and I was thinking something quite quite different: If I have a set that has no elements, then the absence of elements would make the task of assigning a property to one of it's elements impossible. Is that feasible?
 A: Because there are no elements to witness otherwise. This is called vacuous truth in mathematics.
Statements of the form "For every $x$ ..." are false if and only if there is a counterexample. The statement "For every $x$, if $x\in\varnothing$ then $x$ is even" has no counterexamples.
A: Each element of the empty set is even can be paraphrased as if $x$ is an element of the empty set, then $x$ is even:
$$\forall x\Big(x\in\varnothing\to x\text{ is even}\Big)\;.\tag{1}$$
How could you show that this was false? You’d have to show that there was some $x\in\varnothing$ that was not even. And you can’t do this: you can’t find any $x$ in the empty set, let alone one that is even. Since you can’t show that $(1)$ is false, it must be true.
To restate the argument in slightly different terms, the statement 

if $x$ is an element of the empty set, then $x$ is even

imposes a condition on elements of the empty set, but the empty set has no elements, so it doesn’t actually impose a condition on anything. Thus, nothing can violate it: no object is an element of the empty set, so no object is even a candidate to violate the requirement of being even.
The usual terminology is that the statement $(1)$ is vacuously true: it’s true because it doesn’t actually impose a requirement on anything. Note that you could replace $x\text{ is even}$ in $(1)$ with pretty much any statement about $x$, and the resulting sentence would be vacuously true by essentially the same argument.
