Subset
In mathematics, especially in set theory, a set A is a subset of a set B, if A is "contained" inside B. Every set is a subset of itself.
More formally, If A and B are sets and every element of A is also an element of B, then:
- A is a subset of (or is included in) B, denoted by A ⊆ B,
or equivalently
- B is a superset of (or includes) A, denoted by B ⊇ A.
If A is a subset of B, but A is not equal to B, then A is also a proper (or strict) subset of B. This is written as A ⊂ B. In the same way, A ⊃ B means that A is a proper superset of B.
An easy way to remember the difference in symbols is to note that ⊆ and ⊂ are analagous to ≤ and <. For example, if A is a subset of B (written as A ⊆ B), then the number of elements in A is less than or equal to the number of elements in B (written as |A| ≤ |B|). Likewise, if A ⊂ B, then |A| < |B|.
Notational variations
There are two major systems in use for the notation of subsets. The older system uses the symbol "⊂" and "⊃" to indicate subsets and supersets while using "⊊" and "" to indicate proper subsets and proper supersets. The newer system is described above. The newer system can be handled by a wider variety of web browsers and lends itself more appropriately to analogies with < and ≤ as described above. Wikipedia uses the newer system.
Examples
- The set {1, 2} is a proper subset of {1, 2, 3}.
- The set of natural numbers is a proper subset of the set of rational numbers.
- The set {x : x is a prime number greater than 2000} is a proper subset of {x : x is an odd number greater than 1000}
- Any set is a subset of itself, but not a proper subset.
- The empty set, written ø, is also a subset of any given set X. (This statement is vacuously true, see proof below) The empty set is always a proper subset, except of itself.
Properties
PROPOSITION 1: The empty set is a subset of every set.
Proof: Given any set A, we wish to prove that ø is a subset of A. This involves showing that all elements of ø are elements of A. But there are no elements of ø.
For the experienced mathematician, the inference " ø has no elements, so all elements of ø are elements of A" is immediate, but it may be more troublesome for the beginner. Since ø 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 ø was not a subset of A, we would have to find an element of ø which was not also an element of A. Since there are no elements of ø, this is impossible and hence ø 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 for unions and intersections (see The fundamental laws of set algebra), show that it is a Boolean algebra.
PROPOSITION 3: If A, B and C are subsets of a set S then the following hold:
- existence of a least element and a greatest element:
- ø ⊆ A ⊆ S (that ø ⊆ 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 unions, intersections and complements.
PROPOSITION 4: For any two sets A and B, the following are equivalent:
- A ⊆ B
- A ∩ B = A
- A ∪ B = B
- A − B = ø
- B′ ⊆ A′
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.
Other properties of inclusion
The usual order on the ordinal numbers is given by inclusion.
For the power set of a set S, the inclusion partial order is (up to an order-isomorphism) the Cartesian product of |S| (the cardinality of S) copies of the partial order on {0,1}, for which 0 < 1.cs:Podmnožina et:Alamhulk es:Subconjunto fr:Sous-ensemble ko:부분집합 is:Hlutmengi it:Sottoinsieme he:תת קבוצה nl:Deelverzameling ja:部分集合 pl:Podzbiór ru:Подмножество sl:Podmnožica uk:Підмножина zh:子集