algebra of sets

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In 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 of
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
, intersection, 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 elit ...
and set
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...
. 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 i ...
with the join operator being ''union'', the meet operator being ''intersection'', the complement operator being ''set complement'', the bottom being $\varnothing$ and the top being the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. ...
set under consideration.

# Fundamentals

The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative 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, and for a full rigorous axiomatic treatment see axiomatic set theory.

# The fundamental properties of set algebra

The binary operations of set
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
($\cup$) and intersection ($\cap$) 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 ...
: ::*$A \cup B = B \cup A$ ::*$A \cap B = B \cap A$ :
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 ...
: ::*$\left(A \cup B\right) \cup C = A \cup \left(B \cup C\right)$ ::*$\left(A \cap B\right) \cap C = A \cap \left(B \cap C\right)$ :
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 arithmet ...
: ::*$A \cup \left(B \cap C\right) = \left(A \cup B\right) \cap \left(A \cup C\right)$ ::*$A \cap \left(B \cup C\right) = \left(A \cap B\right) \cup \left(A \cap C\right)$ 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 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-clas ...
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 : ::*$A \cup \varnothing = A$ ::*$A \cap U = A$ :Complement : ::*$A \cup A^C = U$ ::*$A \cap A^C = \varnothing$ 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 su ...
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 is ...
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 operation o ...
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 $\left(A^C\right)^C = A$, then this is exactly the algebra of propositional linear logic.

# 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: ::*$A \cup A = A$ ::*$A \cap A = A$ :domination laws: ::*$A \cup U = U$ ::*$A \cap \varnothing = \varnothing$ :
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'' ¤ ''b ...
s: ::*$A \cup \left(A \cap B\right) = A$ ::*$A \cap \left(A \cup B\right) = A$ 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 : $A \cap B = A \setminus \left(A \setminus B\right)$

# 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 math ...
: ::*$\left(A \cup B\right)^C = A^C \cap B^C$ ::*$\left(A \cap B\right)^C = A^C \cup B^C$ :double complement or
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
law: ::*$^ = A$ :complement laws for the universe set and the empty set: ::*$\varnothing^C = U$ ::*$U^C = \varnothing$ 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 $A \cup B = U$, and $A \cap B = \varnothing$, then $B = A^C$

# The algebra of inclusion

The following proposition says that
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...
, that is the binary relation 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 bina ...
. PROPOSITION 6: If ''A'', ''B'' and ''C'' are sets then the following hold: : reflexivity: ::*$A \subseteq A$ :
antisymmetry In linguistics, antisymmetry is a syntactic theory presented in Richard S. Kayne's 1994 monograph ''The Antisymmetry of Syntax''. It asserts that grammatical hierarchies in natural language follow a universal order, namely specifier-head-comple ...
: ::*$A \subseteq B$ and $B \subseteq A$ if and only if $A = B$ : transitivity: ::*If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$ 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 post ...
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 i ...
. PROPOSITION 7: If ''A'', ''B'' and ''C'' are subsets of a set ''S'' then the following hold: :existence of a
least element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
and a
greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
: ::*$\varnothing \subseteq A \subseteq S$ :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 topo ...
: ::*$A \subseteq A \cup B$ ::*If $A \subseteq C$ and $B \subseteq C$, then $A \cup B \subseteq C$ :existence of meets: ::*$A \cap B \subseteq A$ ::*If $C \subseteq A$ and $C \subseteq B$, then $C \subseteq A \cap B$ The following proposition says that the statement $A \subseteq B$ is equivalent to various other statements involving unions, intersections and complements. PROPOSITION 8: For any two sets ''A'' and ''B'', the following are equivalent: :*$A \subseteq B$ :*$A \cap B = A$ :*$A \cup B = B$ :*$A \setminus B = \varnothing$ :*$B^C \subseteq A^C$ 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: :*$C \setminus \left(A \cap B\right) = \left(C \setminus A\right) \cup \left(C \setminus B\right)$ :*$C \setminus \left(A \cup B\right) = \left(C \setminus A\right) \cap \left(C \setminus B\right)$ :*$C \setminus \left(B \setminus A\right) = \left(A \cap C\right)\cup\left(C \setminus B\right)$ :*$\left(B \setminus A\right) \cap C = \left(B \cap C\right) \setminus A = B \cap \left(C \setminus A\right)$ :*$\left(B \setminus A\right) \cup C = \left(B \cup C\right) \setminus \left(A \setminus C\right)$ :*$\left(B \setminus A\right) \setminus C = B \setminus \left(A \cup C\right)$ :*$A \setminus A = \varnothing$ :*$\varnothing \setminus A = \varnothing$ :*$A \setminus \varnothing = A$ :*$B \setminus A = A^C \cap B$ :*$\left(B \setminus A\right)^C = A \cup B^C$ :*$U \setminus A = A^C$ :*$A \setminus U = \varnothing$

*
σ-algebra In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
is an algebra of sets, completed to include countably infinite operations. * Axiomatic set theory * 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 un ...
*
List of set identities and relations This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for ...
* Naive set theory *
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, or ...
*
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 po ...
— a subset of $\wp\left(X\right)$, the power set of $X$, closed with respect to arbitrary union, finite intersection and containing $\emptyset$ 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"