Examples and counterexamples in mathematics/Sets

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Set without membersEdit

The empty set, denoted by   (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   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,  . Also,   The latter fact has a set-theoretic counterpart, see the next item.

The powerset of the empty set is not emptyEdit

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   then its power set contains   and nothing else; it is   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   elements. In particular,