# Examples and counterexamples in mathematics/Sets

The empty set, denoted by ${\displaystyle \emptyset }$ (or sometimes ${\displaystyle \{\}}$) 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 ${\displaystyle \emptyset }$ 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, ${\displaystyle \cos 0=1}$. Also, ${\displaystyle 2^{0}=1.}$ The latter fact has a set-theoretic counterpart, see the next item.
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 ${\displaystyle S=\emptyset ,}$ then its power set contains ${\displaystyle \emptyset }$ and nothing else; it is ${\displaystyle \{\emptyset \},}$ that is, ${\displaystyle \{\{\}\}.}$ 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 ${\displaystyle 2^{n}}$ elements. In particular, ${\displaystyle 2^{0}=1.}$