View source for Examples and counterexamples in mathematics/Sets

{{Wikipedia-inline|Set}}

==Set without members==
{{Wikipedia-inline|Empty set}}

The empty set, denoted by <math>\emptyset</math> (or sometimes <math>\{\}</math>) 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 <math>\emptyset</math> 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, <math>\cos 0 = 1</math>. Also, <math>2^0=1.</math> The latter fact has a set-theoretic counterpart, see the next item.

==The powerset of the empty set is not empty==
{{Wikipedia-inline|Power set}}
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 <math>S=\emptyset,</math> then its power set contains <math>\emptyset</math> and nothing else; it is <math>\{\emptyset\},</math> that is, <math>\{\{\}\}.</math> 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 <math>2^n</math> elements. In particular, <math>2^0=1.</math>

{{BookCat}}