• Set at Wikipedia.

• Empty set at Wikipedia.

The empty set, denoted by${\displaystyle \mathrm{\varnothing }}$ (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${\displaystyle \mathrm{\varnothing }}$ 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 functionf we get another numberf(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.

• Power set at Wikipedia.

The power set (or "powerset") of any setS is the set of all subsets ofS, including the empty set andS itself. If${\displaystyle S=\mathrm{\varnothing },}$ then its power set contains${\displaystyle \mathrm{\varnothing }}$ and nothing else; it is${\displaystyle \{\mathrm{\varnothing }\},}$ 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, ifS containsn elements, then its power set contains${\displaystyle 2^n}$ elements. In particular,${\displaystyle 2^0=1.}$

• Book:Examples and counterexamples in mathematics