Subset

A portion of a Set. $B$ is a subset of $A$ (written $B\subseteq A$) Iff every member of $B$ is a member of $A$. If $B$ is a Proper Subset of $A$ (i.e., a subset other than the set itself), this is written $B\subset A$.

A Set of $n$ elements has $2^n$ subsets (including the set itself and the Empty Set). For sets of $n=1$, 2, ... elements, the numbers of subsets are therefore 2, 4, 8, 16, 32, 64, ... (Sloane's A000079). For example, the set $\{1\}$ has the two subsets $\emptyset$ and $\{1\}$. Similarly, the set $\{1,2\}$ has subsets $\emptyset$ (the Empty Set, $\{1\}$, $\{2\}$, and $\{1,2\}$.

See also Empty Set, Implies, k-Subset, Proper Subset, Superset, Venn Diagram

References

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 109, 1996.

Ruskey, F. ``Information of Subsets of a Set.'' http://sue.csc.uvic.ca/~cos/inf/comb/SubsetInfo.html.

Sloane, N. J. A. Sequence A000079/M1129 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.