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.)
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 .