Algebra of sets
Encyclopedia
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.
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 the article on sets, for a fuller account see naive set theory
, and for a full rigorous axiom
atic treatment see axiomatic set theory.
s 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:
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.
The next proposition, states two additional pairs of laws involving three specials sets: the empty set
, the universal set
and the complement
of a set.
PROPOSITION 2: For any subset
A of universal set U, where Ø is the empty set, the following identities hold:
The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity element
s for union and intersection, respectively.
Unlike addition and multiplication, union and intersection do not have inverse element
s. 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.
These are examples of an extremely important and powerful property of set algebra, namely, the principle 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 Ø and reversing inclusions is also true. A statement is said to be self-dual if it is equal to its own dual.
PROPOSITION 3: For any subset
s A and B of a universal set U, the following identities hold:
As noted above each of the laws stated in proposition 3, can be derived from the five fundamental pairs of laws stated in proposition 1 and proposition 2. As an illustration, a proof is given below for the idempotent law for union.
Proof:
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.
Proof:
Intersection can be expressed in terms of union and set difference :
PROPOSITION 4: Let A and B be subset
s of a universe U, then:
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:
PROPOSITION 6: If A, B and C are sets then the following hold:
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:
The following proposition says that the statement
is equivalent to various other statements involving unions, intersections and complements.
PROPOSITION 8: 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 operations of set union or set intersection, which means that the notion of set inclusion is axiomatically superfluous.
or set-theoretic difference.
PROPOSITION 9: For any universe U and subsets A, B, and C of U, the following identities hold:
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
, intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
, and complementation
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
and the relations
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
of set equality and set inclusion
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Introduction
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
Elementary algebra
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...
that develops the fundamental properties of arithmetic operations and relations.
For example, the operations of addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
and multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
obey familiar laws such as associativity
Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
, commutativity
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
and distributivity
Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
; while the "less than or equal" relation satisfies such laws as reflexivity
Reflexive relation
In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
, antisymmetry
Antisymmetric relation
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
and transitivity
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
. 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 the article on sets, for a fuller account see naive set theory
Naive set theory
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
, and for a full rigorous axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
atic treatment see axiomatic set theory.
The fundamental laws of set algebra
The binary operationBinary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
s of set union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
and intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
satisfy many identities
Identity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
. Several of these identities or "laws" have well established names. Three pairs of laws, are stated, without proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
, in the following proposition.
PROPOSITION 1: For any sets A, B, and C, the following identities hold:
- commutative laws:
-
- associativeAssociativityIn mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
laws: -
- distributiveDistributivityIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
laws: -
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.
The next proposition, states two additional pairs of laws involving three specials sets: the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
, the universal set
Universal set
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, the conception of a set of all sets leads to a paradox...
and the complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of a set.
PROPOSITION 2: For any subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
A of universal set U, where Ø is the empty set, the following identities hold:
- identity laws:
-
- complement laws:
-
The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
s for union and intersection, respectively.
Unlike addition and multiplication, union and intersection do not have inverse element
Inverse element
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
s. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation
Unary operation
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation on A....
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 Ø and U.These are examples of an extremely important and powerful property of set algebra, namely, the principle 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 Ø 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 subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s A and B of a universal set U, the following identities hold:
- idempotent laws:
-
- domination laws:
-
- absorption lawsAbsorption lawIn algebra, the absorption law or absorption identity is an identity linking a pair of binary operations.Two binary operations, say ¤ and *, are said to be connected by the absorption law if:...
: -
As noted above each of the laws stated in proposition 3, can be derived from the five fundamental pairs of laws stated in proposition 1 and proposition 2. As an illustration, a proof is given below for the idempotent law for union.
Proof:
 |
 |
by the identity law of intersection |
 |
by the complement law for union | |
 |
by the distributive law of union over intersection | |
 |
by the complement law for intersection | |
 |
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.
Proof:
 |
 |
by the identity law for union |
 |
by the complement law for intersection | |
 |
by the distributive law of intersection over union | |
 |
by the complement law for union | |
 |
by the identity law for intersection |
Intersection can be expressed in terms of union and set difference :
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 subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of a universe U, then:
- De Morgan's laws:
-
- double complement or Involution law:
-
- complement laws for the universal set and the empty set:
-
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
, and 
, then 
- If
-
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:
- reflexivityReflexive relationIn mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
: -
- antisymmetryAntisymmetric relationIn mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
:-
and 
if and only if 
-
- transitivityTransitive relationIn mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
:-
- If
and 
, then 
- If
-
The following proposition says that for any set S, the power set of S, ordered by inclusion, is a bounded lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
, 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 a least elementGreatest 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...
and a greatest elementGreatest 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...
: -
- existence of joinsLattice (order)In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
:-
- If
and 
, then 
-
- existence of meetsLattice (order)In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
:-
- If
and 
, then 
-
The following proposition says that the statement
is equivalent to various other statements involving unions, intersections and complements.PROPOSITION 8: 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 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 complementsComplement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
or set-theoretic difference.
PROPOSITION 9: For any universe U and subsets A, B, and C of U, the following identities hold:
See also
- Set (mathematics)
- Field of sets
- Naive set theoryNaive set theoryNaive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
- Axiomatic set theory