Empty relation on non-empty set - is it reflexive? I have here a relation defined by a conjunction of contradictory conditions, i.e. 
 $R = \{ \langle x,y \rangle \in \mathbb N ^2  |  x = y^2 \land y = -x \}$
So, at least as I understand my undergrad course so far, this produces an empty relation, since there are no pairs of natural numbers such that one is the negative of the other.
I can see why such a relation would be symmetrical and transitive, since both of these properties involve an implication.
However, I am trying to determine if this relation is reflexive or irreflexive. My professor suggested that this would be computable for any relation, but in my mind, reflexivity is defined based on the presence of at least one element in the relation's domain. And if there are no elements in the relation, then how can they be in relation with themselves?
Some help, any help, would be appreciated!!
P.S. If anybody can recommend a modern manual for this subject, I'd really appreciate it. There is no recommended reading in this course, and the classes aren't very enlightening. I work better from a book.
 A: Assuming your definition of $\Bbb N$ excludes $0$ (since some do, some don't), then you are correct in concluding that $R = \varnothing$, since both equations are satisfied only for $(x,y) \in \{(0,0),(1,-1)\}$. You can prove this algebraically, or use the fact that $\Bbb N$ contains no negatives, of course, but the graph makes it especially clear in my opinion:

The question of whether the relation $R = \varnothing$ is reflexive is slightly complicated though. In the case the relation is over a nonempty set, then it is easy to verify that it is not reflexive, since no element of that nonempty set is related to anything, especially not itself. (For instance, $(1,1) \not \in R = \varnothing \subseteq \Bbb N^2$ in your case.)
However, in the hypothetical that we're defining the relation $R$ over the empty set, then the empty relation is reflexive. Namely, this is because there exists no $x \in \varnothing$ for which $(x,x) \not \in R = \varnothing$ -- i.e. vacuous logic.
Curiously, you can also see that while relations over a nonempty set are always reflexive, irreflexive, or neither, (empty) relations over the empty set are both reflexive and irreflexive.
While not strictly related to your example, I figured it would be worth noting both cases (nonempty and empty sets) for completeness's sake. Of course, your example is simply not reflexive, since $\Bbb N^2 \ne \varnothing$.

...granted, this question is quite old, so I imagine you don't need help now. But hopefully this helps someone in the future, and, if nothing else, gets this question out of the unanswered queue.
