A subset is a portion of a set. B is a subset of A (written B subset= 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. If B is not a subset of A, this is written B !subset= A. (The notation B !subset A is generally not used, since B !subset= A automatically means that B and A 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 S is called the power set of S, and a set of n elements has 2^n 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 n elements is given by the binomial sum

 sum_(k=0)^n(n; k)=2^n.

For sets of n=1, 2, ... elements, the numbers of subsets are therefore 2, 4, 8, 16, 32, 64, ... (OEIS 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, Improper Subset, k-Subset, p-System, Power Set, Proper Subset, Subset Sum Problem, Superset, Venn Diagram Explore this topic in the MathWorld classroom

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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., E. Theory of Sets. New York: Dover, p. 6, 1950.Ruskey, F. "Information of Subsets of a Set.", 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."

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Cite this as:

Weisstein, Eric W. "Subset." From MathWorld--A Wolfram Web Resource.

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