I mean, if we want to run with the man analogies... The empty set is a valid set. Perfectly reasonable, and completely necessary in many cases. But the empty

I mean, if we want to run with the man analogies... The empty set is a valid set. Perfectly reasonable, and completely necessary in many cases. But the empty Group is not a valid Group. A Group requires an identity element, which means non-empty. The smallest valid Group, the Trivial Group, still requires an element. I would argue a diploma or a banknote resemble a Group more than a Set. There are required elements, such as an issuing institution.