Examples and counterexamples in mathematics/Sets

• Set at Wikipedia.

Set without members

• Empty set at Wikipedia.

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

• Power set at Wikipedia.

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, $2^{0}=1.$