Subset
If X and Y are sets and every element of X is also an element of Y, then we say or write:
- X is a subset of (or is included in) Y;
- X ⊆ Y;
- Y is a superset of (or includes) X;
- Y ⊇ X.
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.
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. AskFactMaster.Com uses the newer system, which can be handled by a wider variety of web browsers. Analogous comments apply to supersets.
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 ∅, 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 ∅ 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 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:
- ∅ ⊆ 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 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 = ∅
- 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.
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Categories: Set theory