How this calculator works
A is a subset of B when every element in A also belongs to B. It is proper when the sets are unequal as well. Set equality ignores order and repeated entries. Enumeration chooses exactly k distinct elements of A and lists each choice once. The empty subset occurs when k is zero. Cardinality-only mode counts without listing objects and uses exact integers up to n = 1000. Both proper-subset count and nonempty-subset count equal 2^n - 1, although they exclude different subsets.
Formula
A subseteq B means every x in A is in B. A proper subset additionally requires A != B. The number of k-element subsets is C(n,k).
Worked example
{1, 2} is a proper subset of {1, 2, 3}. The size-2 subsets of {1, 2, 3} are {1, 2}, {1, 3} and {2, 3}, so the count is 3.
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Frequently asked questions
Is the empty set a subset?
Yes. It has no element that can violate membership. It is a proper subset of every nonempty set.
Does order matter?
No. {1, 2} and {2, 1} describe the same set.
Can I enumerate k larger than the set size?
This tool rejects that request. Choose a whole-number size from zero through the distinct element count.