Subset

A subset is a portion of a set.  is a subset of  (written ) iff every member of  is a member of . If  is a proper subset of  (i.e., a subset other than the set itself), this is written . If  is not a subset of , this is written . (The notation  is generally not used, since  automatically means that  and  cannot be the same.)

The subsets (i.e., power set) of a given set can be found using Subsets[list].

An efficient algorithm for obtaining the next higher number having the same number of 1 bits as a given number (which corresponds to computing the next subset) is given by Gosper (1972) in PDP-10 assembler.

The set of subsets of a set  is called the power set of , and a set of  elements has  subsets (including both the set itself and the empty set). This follows from the fact that the total number of distinct k-subsets on a set of  elements is given by the binomial sum

For sets of , 2, ... elements, the numbers of subsets are therefore 2, 4, 8, 16, 32, 64, ... (OEIS A000079). For example, the set  has the two subsets  and . Similarly, the set  has subsets  (the empty set), , , and .

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.

Gosper, R. W. Item 175 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/hacks.html#item175.

Kamke, E. Theory of Sets. New York: Dover, p. 6, 1950.

Ruskey, F. "Information of Subsets of a Set." http://www.theory.csc.uvic.ca/~cos/inf/comb/SubsetInfo.html.Skiena, S. "Binary Representation and Random Sets." §1.5.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 41-42, 1990.

Sloane, N. J. A. Sequence A000079/M1129 in "The On-Line Encyclopedia of Integer Sequences."