The algebra of sets
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The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
The algebra of sets is the development of the fundamental properties of set operations and set relations. These properties provide insight into the fundamental nature of sets. They also have practical considerations.
Just like expressions and calculations in ordinary arithmetic, expressions and calculations involving sets can be quite complex. It is helpful to have systematic procedures available for manipulating and evaluating such expressions and performing such computations.
In the case of arithmetic, it is elementary algebra that develops the fundamental properties of arithmetic operations and relations.
For example, the operations of addition and multiplication obey familiar laws such as associativity, commutativity and distributivity, while, the "less than or equal" relation satisfies such laws as reflexivity, antisymmetry and transitivity. These laws provide tools which facilitate computation, as well as describe the fundamental nature of numbers, their operations and relations.
The algebra of sets is the set-theoretic analogue of the algebra of numbers. It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. These are the topics covered in this article. For a basic introduction to sets see, Set, for a fuller account see Naive set theory.
The fundamental laws of set algebra
The binary operations of set union and intersection satisfy many identities. Several of these identities or "laws" have well established names. Three pairs of laws, are stated, without proof, in the following proposition.
PROPOSITION 1: For any sets A, B, and C, the following identities hold:
- commutative laws:
- A ∪ B = B ∪ A
- A ∩ B = B ∩ A
- associative laws:
- (A ∪ B ) ∪ C = A ∪ (B ∪ C )
- (A ∩ B ) ∩ C = A ∩ (B ∩ C )
- distributive laws:
- A ∪ (B ∩ C ) = (A ∪ B ) ∩ (A ∪ C )
- A ∩ (B ∪ C ) = (A ∩ B ) ∪ (A ∩ C )
Notice that the analogy between unions and intersections of sets, and addition and multiplication of numbers, is quite striking. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection distributes over unions. However, unlike addition and multiplication, union also distributes over intersection.
PROPOSITION 2: For any subset A of universal set U, the following identities hold:
- identity laws:
- A ∪ ∅ = A
- A ∩ U = A
- complement laws:
- A ∪ A′ = U
- A ∩ A′ = ∅
The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, ∅ and U are the identity elements for union and intersection, respectively.
Unlike addition and multiplication, union and intersection do not have inverse elements. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation of set complementation.
The preceding five pairs of laws, the commutative, associative, distributive, identity and complement laws can be said to encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.
The principle of duality
The above propositions display the following interesting pattern. Each of the identities stated above is one of a pair of identities, such that, each can be transformed into the other by interchanging ∪ and ∩, and also ∅ and U.
These are examples of an extremely important and powerful property of set algebra, namely, the principal of duality for sets, which asserts that for any true statement about sets, the dual statement obtained by interchanging unions and intersections, interchanging U and ∅ and reversing inclusions is also true. A statement is said to be self-dual if it is equal to its own dual.
Some additional laws for unions and intersections
The following proposition states six more important laws of set algebra, involving unions and intersections.
PROPOSITION 3: For any subsets A and B of a universal set U, the following identities hold:
- idempotent laws:
- A ∪ A = A
- A ∩ A = A
- domination laws:
- A ∪ U = U
- A ∩ ∅ = ∅
- absorption laws:
- A ∪ (A ∩ B ) = A
- A ∩ (A ∪ B ) = A
As noted above each of the laws stated in proposition 3, can be derived from the five fundamental laws stated in proposition 1 and proposition 2. As an illustration, a proof is given below for the idempotent law for union.
|A ∪ A||= (A ∪ A) ∩ U||by the identity law for intersection|
|= (A ∪ A) ∩ (A ∪ A′)||by the complement law for union|
|= A ∪ (A ∩ A′)||by the distributive law of union over intersection|
|= A ∪ ∅||by the complement law for intersection|
|= A||by the identity law for union|
The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law for union, namely the idempotent law for intersection.
|A ∩ A||= (A ∩ A) ∪ ∅||by the identity law for union|
|= (A ∩ A) ∪ (A ∩ A′)||by the complement law for intersection|
|= A ∩ (A ∪ A′)||by the distributive law of intersection over union|
|= A ∩ U||by the complement law for union|
|= A||by the identity law for intersection|
Some additional laws for complements
The following proposition states five more important laws of set algebra, involving complements.
PROPOSITION 4: Let A and B be subsets of a universe U, then:
- De Morgan's laws:
- (A ∪ B )′ = A′ ∩ B′
- (A ∩ B )′ = A′ ∪ B′
- double complement or Involution law:
- A′′ = A
- complement laws for the universal set and the empty set:
- ∅′ = U
- U′ = ∅
Notice that the double complement law is self-dual.
The next proposition, which is also self-dual, says that the complement of a set is the only set that satisfies the complement laws. In other words, complementation is characterized by the complement laws.
PROPOSITION 5: Let A and B be subsets of a universe U, then:
- uniqueness of complements:
- If A ∪ B = U, and A ∩ B = ∅ then B = A′.
The algebra of inclusion
The following proposition says that inclusion is a partial order.
PROPOSITION 6: If A, B and C are sets then the following hold:
- A ⊆ A
- A ⊆ B and B ⊆ A if and only if A = B
- 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 7: If A, B and C are subsets of a set S then the following hold:
- 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 8: 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 operations of set union or set intersection, which means that the notion of set inclusion is axiomatically superfluous.
The algebra of relative complements
The following proposition, lists several identities concerning relative complements or set-theoretic difference.
PROPOSITION 9: For any universe U and subsets A, B, and C of U, the following identities hold:
- C − (A ∩ B ) = (C − A ) ∪ (C − B )
- C − (A ∪ B ) = (C − A ) ∩ (C − B )
- C − (B − A ) = (A ∩ C ) ∪ (C − B )
- (B − A ) ∩ C = (B ∩ C ) − A = B ∩ (C − A )
- (B − A ) ∪ C = (B ∪ C ) − (A − C )
- A − A = ∅
- ∅ − A = ∅
- A − ∅ = A
- B − A = A' ∩ B
- (B − A )' = A ∪ B′
- U − A = A′
- A − U = ∅