In

"The Algebra of Sets", pp 16—23

* Courant, Richard, Herbert Robbins, Ian Stewart, ''What is mathematics?: An Elementary Approach to Ideas and Methods'', Oxford University Press US, 1996.

"SUPPLEMENT TO CHAPTER II THE ALGEBRA OF SETS"

{{Mathematical logic Basic concepts in set theory

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Ma ...

of union, intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...

, and complementation and the relations of set equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elite ...

and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Any set of sets closed under the set-theoretic operations forms a Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...

with the join operator being ''union'', the meet operator being ''intersection'', the complement operator being ''set complement'', the bottom being $\backslash varnothing$ and the top being the universe set under consideration.
Fundamentals

The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmeticaddition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...

and multiplication are associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...

and commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It d ...

, and for a full rigorous axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

atic treatment see axiomatic set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

.
The fundamental properties of set algebra

Thebinary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary ope ...

s of set union ($\backslash cup$) and intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...

($\backslash cap$) satisfy many identities. Several of these identities or "laws" have well established names.
:Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

:
::*$A\; \backslash cup\; B\; =\; B\; \backslash cup\; A$
::*$A\; \backslash cap\; B\; =\; B\; \backslash cap\; A$
:Associative property
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...

:
::*$(A\; \backslash cup\; B)\; \backslash cup\; C\; =\; A\; \backslash cup\; (B\; \backslash cup\; C)$
::*$(A\; \backslash cap\; B)\; \backslash cap\; C\; =\; A\; \backslash cap\; (B\; \backslash cap\; C)$
:Distributive property
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...

:
::*$A\; \backslash cup\; (B\; \backslash cap\; C)\; =\; (A\; \backslash cup\; B)\; \backslash cap\; (A\; \backslash cup\; C)$
::*$A\; \backslash cap\; (B\; \backslash cup\; C)\; =\; (A\; \backslash cap\; B)\; \backslash cup\; (A\; \backslash cap\; C)$
The union and intersection of sets may be seen as analogous to the addition and multiplication of numbers. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection ''distributes'' over union. However, unlike addition and multiplication, union also distributes over intersection.
Two additional pairs of properties involve the special sets called the empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

Ø and the universe set $U$; together with the complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...

operator ($A^C$ denotes the complement of $A$. This can also be written as $A\text{'}$, read as A prime). The empty set has no members, and the universe set has all possible members (in a particular context).
:Identity :
::*$A\; \backslash cup\; \backslash varnothing\; =\; A$
::*$A\; \backslash cap\; U\; =\; A$
:Complement :
::*$A\; \backslash cup\; A^C\; =\; U$
::*$A\; \backslash cap\; A^C\; =\; \backslash varnothing$
The identity expressions (together with the commutative expressions) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...

s for union and intersection, respectively.
Unlike addition and multiplication, union and intersection do not have inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that i ...

s. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operatio ...

of set complementation.
The preceding five pairs of formulae—the commutative, associative, distributive, identity and complement formulae—encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.
Note that if the complement formulae are weakened to the rule $(A^C)^C\; =\; A$, then this is exactly the algebra of propositional linear logic
Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also ...

.
The principle of duality

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 subsets ''A'' and ''B'' of a universe set U, the following identities hold: :idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

laws:
::*$A\; \backslash cup\; A\; =\; A$
::*$A\; \backslash cap\; A\; =\; A$
:domination laws:
::*$A\; \backslash cup\; U\; =\; U$
::*$A\; \backslash cap\; \backslash varnothing\; =\; \backslash varnothing$
:absorption law
In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations.
Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if:
:''a'' ¤ (''a'' ⁂ ''b'') = ''a'' ⁂ (''a'' ¤ ' ...

s:
::*$A\; \backslash cup\; (A\; \backslash cap\; B)\; =\; A$
::*$A\; \backslash cap\; (A\; \backslash cup\; B)\; =\; A$
As noted above, each of the laws stated in proposition 3 can be derived from the five fundamental pairs of laws stated above. 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 set difference :
$A\; \backslash cap\; B\; =\; A\; \backslash setminus\; (A\; \backslash setminus\; B)$
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
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathe ...

:
::*$(A\; \backslash cup\; B)^C\; =\; A^C\; \backslash cap\; B^C$
::*$(A\; \backslash cap\; B)^C\; =\; A^C\; \backslash cup\; B^C$
:double complement or involution law:
::*$^\; =\; A$
:complement laws for the universe set and the empty set:
::*$\backslash varnothing^C\; =\; U$
::*$U^C\; =\; \backslash varnothing$
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 $A\; \backslash cup\; B\; =\; U$, and $A\; \backslash cap\; B\; =\; \backslash varnothing$, then $B\; =\; A^C$
The algebra of inclusion

The following proposition says that inclusion, that is thebinary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...

of one set being a subset of another, is a partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

.
PROPOSITION 6: If ''A'', ''B'' and ''C'' are sets then the following hold:
: reflexivity:
::*$A\; \backslash subseteq\; A$
: antisymmetry:
::*$A\; \backslash subseteq\; B$ and $B\; \backslash subseteq\; A$ if and only if $A\; =\; B$
: transitivity:
::*If $A\; \backslash subseteq\; B$ and $B\; \backslash subseteq\; C$, then $A\; \backslash subseteq\; C$
The following proposition says that for any set ''S'', the power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is p ...

of ''S'', ordered by inclusion, is a bounded lattice
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper boun ...

, and hence together with the distributive and complement laws above, show that it is a Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...

.
PROPOSITION 7: If ''A'', ''B'' and ''C'' are subsets of a set ''S'' then the following hold:
:existence of a least element and a greatest element:
::*$\backslash varnothing\; \backslash subseteq\; A\; \backslash subseteq\; S$
:existence of joins Join may refer to:
* Join (law), to include additional counts or additional defendants on an indictment
*In mathematics:
** Join (mathematics), a least upper bound of sets orders in lattice theory
** Join (topology), an operation combining two top ...

:
::*$A\; \backslash subseteq\; A\; \backslash cup\; B$
::*If $A\; \backslash subseteq\; C$ and $B\; \backslash subseteq\; C$, then $A\; \backslash cup\; B\; \backslash subseteq\; C$
:existence of meets:
::*$A\; \backslash cap\; B\; \backslash subseteq\; A$
::*If $C\; \backslash subseteq\; A$ and $C\; \backslash subseteq\; B$, then $C\; \backslash subseteq\; A\; \backslash cap\; B$
The following proposition says that the statement $A\; \backslash subseteq\; B$ is equivalent to various other statements involving unions, intersections and complements.
PROPOSITION 8: For any two sets ''A'' and ''B'', the following are equivalent:
:*$A\; \backslash subseteq\; B$
:*$A\; \backslash cap\; B\; =\; A$
:*$A\; \backslash cup\; B\; =\; B$
:*$A\; \backslash setminus\; B\; =\; \backslash varnothing$
:*$B^C\; \backslash subseteq\; A^C$
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 and set-theoretic differences. PROPOSITION 9: For any universe U and subsets ''A'', ''B'', and ''C'' of U, the following identities hold: :*$C\; \backslash setminus\; (A\; \backslash cap\; B)\; =\; (C\; \backslash setminus\; A)\; \backslash cup\; (C\; \backslash setminus\; B)$ :*$C\; \backslash setminus\; (A\; \backslash cup\; B)\; =\; (C\; \backslash setminus\; A)\; \backslash cap\; (C\; \backslash setminus\; B)$ :*$C\; \backslash setminus\; (B\; \backslash setminus\; A)\; =\; (A\; \backslash cap\; C)\backslash cup(C\; \backslash setminus\; B)$ :*$(B\; \backslash setminus\; A)\; \backslash cap\; C\; =\; (B\; \backslash cap\; C)\; \backslash setminus\; A\; =\; B\; \backslash cap\; (C\; \backslash setminus\; A)$ :*$(B\; \backslash setminus\; A)\; \backslash cup\; C\; =\; (B\; \backslash cup\; C)\; \backslash setminus\; (A\; \backslash setminus\; C)$ :*$(B\; \backslash setminus\; A)\; \backslash setminus\; C\; =\; B\; \backslash setminus\; (A\; \backslash cup\; C)$ :*$A\; \backslash setminus\; A\; =\; \backslash varnothing$ :*$\backslash varnothing\; \backslash setminus\; A\; =\; \backslash varnothing$ :*$A\; \backslash setminus\; \backslash varnothing\; =\; A$ :*$B\; \backslash setminus\; A\; =\; A^C\; \backslash cap\; B$ :*$(B\; \backslash setminus\; A)^C\; =\; A\; \backslash cup\; B^C$ :*$U\; \backslash setminus\; A\; =\; A^C$ :*$A\; \backslash setminus\; U\; =\; \backslash varnothing$See also

* σ-algebra is an algebra of sets, completed to include countably infinite operations. *Axiomatic set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

* Image (mathematics)#Properties
* Field of sets
In mathematics, a field of sets is a mathematical structure consisting of a pair ( X, \mathcal ) consisting of a set X and a family \mathcal of subsets of X called an algebra over X that contains the empty set as an element, and is closed unde ...

* List of set identities and relations
* Naive set theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It d ...

* Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, o ...

* Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

— a subset of $\backslash wp(X)$, the power set of $X$, closed with respect to arbitrary union, finite intersection and containing $\backslash emptyset$ and $X$.
References

* Stoll, Robert R.; ''Set Theory and Logic'', Mineola, N.Y.: Dover Publications (1979)"The Algebra of Sets", pp 16—23

* Courant, Richard, Herbert Robbins, Ian Stewart, ''What is mathematics?: An Elementary Approach to Ideas and Methods'', Oxford University Press US, 1996.

"SUPPLEMENT TO CHAPTER II THE ALGEBRA OF SETS"

External links

{{Mathematical logic Basic concepts in set theory