Examples and counterexamples in mathematics/Sets

Set without members
The empty set, denoted by $$\emptyset$$ (or sometimes $$\{\}$$) contains no members. If you find it strange and disturbing, think about the number zero (denoted 0); it was a strange and disturbing idea, but now is generally accepted. The number of members in $$\emptyset$$ is 0.

The empty set is a set, not "absence of set". Likewise, an empty box is a box, not "absence of box"; and 0 is a number, not "absence of number". Substituting 0 into a function f we get another number f(0), generally not 0. For example, $$\cos 0 = 1$$. Also, $$2^0=1.$$ The latter fact has a set-theoretic counterpart, see the next item.

The powerset of the empty set is not empty
The power set (or "powerset") of any set S is the set of all subsets of S, including the empty set and S itself. If $$S=\emptyset,$$ then its power set contains $$\emptyset$$ and nothing else; it is $$\{\emptyset\},$$ that is, $$\{\{\}\}.$$ Likewise a box that contains only an empty box is a non-empty box. The number of elements in this power set is 1. Generally, if S contains n elements, then its power set contains $$2^n$$ elements. In particular, $$2^0=1.$$