TITLE: Proof that the empty set is a relation
QUESTION [23 upvotes]: In the book Naive Set Theory, Halmos mentions that the "The least exciting relation is the empty one." and proves that the empty set is a set of ordered pairs because there is no element of the empty set that is not an ordered pair. Since the empty set is a set of ordered pairs, it follows that it is a relation. 
I understand this line of reasoning but couldn't I use that same line of reasoning to prove that the empty set is a set of singletons? And since the empty set is a set of singletons (because it contains no elements which are not singletons) it is not a relation (because a relation is a set of ordered pairs, not singletons). Why is this reasoning invalid?

REPLY [2 votes]: A set of singletons can be a relationship as long as the set of singletons doesn't actually have any singletons or anything else that isn't an ordered pair.  The empty set is a set of singletons that doesn't have any singletons and doesn't have anything that isn't an ordered pair.  So the empty set is a set of singletons that is a relation.  It's the only set of singletons that is.