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If X and Y are sets and every element of X is also an element of Y, then we say or write:

Every set Y is a subset of itself. A subset of Y which is not equal to Y is called proper (or strict). If X is a proper subset of Y, then we write X ⊂ Y. Analogous comments apply to supersets. The relation "is a subset of" is called inclusion.

1 Notational variations

There are two major systems in use for the notation of subsets. The older system uses the symbol "⊂" to indicate any subset and uses "⊊" to indicate proper subsets. The newer system uses the symbol "⊆" to indicate any subsets and uses "⊂" to indicate proper subsets. Wikipedia uses the newer system, which can be handled by a wider variety of web browsers. Analogous comments apply to supersets.

2 Examples

3 Properties

PROPOSITION 1: The empty set is a subset of every set.

Proof: Given any set A, we wish to prove that ∅ is a subset of A. This involves showing that all elements of ∅ are elements of A. But there are no elements of ∅.

For the experienced mathematician, the inference "∅ has no elements, so all elements of ∅ are elements of A" is immediate, but it may be more troublesome for the beginner. Since ∅ has no members at all, how can "they" be members of anything else? It may help to think of it the other way around. In order to prove that ∅ was not a subset of A, we would have to find an element of ∅ which was not also an element of A. Since there are no elements of ∅, this is impossible and hence ∅ is indeed a subset of A.

The following proposition says that inclusion is a partial order.

PROPOSITION 2: If A, B and C are sets then the following hold:

reflexivity:
  • A ⊆ A
antisymmetry:
  • A ⊆ B and B ⊆ A if and only if A = B
transitivity:
  • If A ⊆ B and B ⊆ C then A ⊆ C

The following proposition says that for any set S the power set of S ordered by inclusion is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebraIn mathematics and computer science, Boolean algebras or Boolean lattices are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and compleme.

PROPOSITION 3: If A, B and C are subsets of a set S then the following hold:

existence of a least elementIn mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S''. The term least element is defined dually. Formally, given a partially and a greatest elementIn mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S''. The term least element is defined dually. Formally, given a partially:
  • ∅ ⊆ A ⊆ S (that ∅ ⊆ A is Proposition 1 above.)
existence of joins:
  • A ⊆ A∪B
  • If A ⊆ C and B ⊆ C then A∪B ⊆ C
existence of meets:
  • A∩B ⊆ A
  • If C ⊆ A and C ⊆ B then C ⊆ A∩B

The following proposition says that, the statement "A ⊆ B ", is equivalent to various other statements involving unionsAbstract algebra Algebra In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. This article uses mathematical symbols. Basic definition If, intersectionsIn mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. This article uses mathematical symbols. The intersecti and complements.

PROPOSITION 4: For any two sets A and B, the following are equivalent:

The above proposition shows that the relation of set inclusion can be characterized by either of the set operations of union or intersection, which means that the notion of set inclusion is axiomatically superfluous.







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Subset Set Empty Proposition Proper Elements

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