In the field of set theory a superset is a set that contains all the elements of another set. It is the opposite of subset.

More formally, set A is a superset of set B iff all elements contained in B are also contained in A. This would be written AB.

For example, if A = {a,b,c,d} and B = {a,c}, then A is the superset of B.

Interesting things to note about superset:

  • Every set is the superset of itself. (A contains everything A contains, doesn't it?)
  • Every set is the superset of the empty set. This is sometimes called the null set or written ∅ (∅ in HTML).

See also: proper superset, subset, subset (math).

Technically, the correct HTML symbol for 'superset' (or, more fully, 'improper superset') is ⊇ (⊇). (AB), or "A is an improper superset of B", means (∀ x : (xB) ⇒ (xA)). The ⊃ (⊃) symbol is for 'proper superset'. AB, or "A is a proper superset of B", means ((AB) ∧ (AB)), or equivalently, ((∀ x : (xB) ⇒ (xA)) ∧ (∃ y : ((yA) ∧ (yB)))). An improper superset of S contains every element of S and may or may not contain more elements, a proper superset of S contains every element of S and at least one more. The term 'superset' (without a qualifier) means either type, which really means the same thing as improper superset (since every proper superset is also an improper one).

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