Set (mathematics)
In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. The arrangement of the objects in the set does not matter. A set may be denoted by placing its objects between a pair of curly braces. For example, the numbers 2, 4, and 6 are distinct objects when considered separately; when considered collectively, they form a single set of size three, written as, which could also be written as,,, or.
The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, the theory of sets is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived.
Etymology
The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite.Definition
A set is a well-defined collection of distinct objects. The objects that make up a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, one of the founders of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.
For technical reasons, Cantor's definition turned out to be inadequate; today, in contexts where more rigor is required, one can use axiomatic set theory, in which the notion of a "set" is taken as a primitive notion and the properties of sets are defined by a collection of axioms. The most basic properties are that a set can have elements, and that two sets are equal if and only if every element of each set is an element of the other; this property is called the extensionality of sets.
Set notation
There are two common ways of describing, or specifying the members of, a set: roster notation and set builder notation. These are examples of extensional and intensional definitions of sets, respectively.Roster notation
The Roster notation method of defining a set consist of listing each member of the set. More specifically, in roster notation, the set is denoted by enclosing the list of members in curly brackets:For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified in roster notation as
where the ellipsis indicates that the list continues in according to the demonstrated pattern.
In roster notation, listing a member repeatedly does not change the set, for example, the set is identical to the set. Moreover, the order in which the elements of a set are listed is irrelevant, so is yet again the same set.
Set-builder notation
In set-builder notation, the set is specified as a selection from a larger set, determined by a condition involving the elements. For example, a set F can be specified as follows:In this notation, the vertical bar means "such that", and the description can be interpreted as "F is the set of all numbers n, such that n is an integer in the range from 0 to 19 inclusive". Sometimes the colon is used instead of the vertical bar.
Set-builder notation is an example of intensional definition.
Other ways of defining sets
Another method is by using a rule or semantic description:This is another example of intensional definition.
Membership
If B is a set and x is one of the objects of B, this is denoted as x ∈ B, and is read as "x is an element of B", as "x belongs to B", or "x is in B". If y is not a member of B then this is written as y ∉ B, read as "y is not an element of B", or "y is not in B".For example, with respect to the sets A =, B =, and F =,
Subsets
If every element of set A is also in B, then A is said to be a subset of B, written A ⊆ B. Equivalently, one can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B.If A is a subset of B, but not equal to B, then A is called a proper subset of B, written A ⊊ B, or simply A ⊂ B, or B ⊋ A.
The expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B, whereas others use them to mean the same as A ⊊ B.
Examples:
- The set of all humans is a proper subset of the set of all mammals.
- ⊆.
- ⊆.
The empty set is a subset of every set, and every set is a subset of itself:
- ∅ ⊆ A.
- A ⊆ A.
Partitions
Power sets
The power set of a set S is the set of all subsets of S. The power set contains S itself and the empty set because these are both subsets of S. For example, the power set of the set is. The power set of a set S is usually written as P.The power set of a finite set with n elements has 2^{n} elements. For example, the set contains three elements, and the power set shown above contains 2^{3} = 8 elements.
The power set of an infinite set is always uncountable. Moreover, the power set of a set is always strictly "bigger" than the original set in the sense that there is no way to pair every element of S with exactly one element of P.
Cardinality
The cardinality of a set S, denoted, is the number of members of S. For example, if B =, then. Repeated members in roster notation are not counted, so, too.The cardinality of the empty set is zero.
Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.
Special sets
There are some sets or kinds of sets that hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the empty set, denoted or ∅. A set with exactly one element, x, is a unit set, or singleton, ; the latter is usually distinct from x.Many of these sets are represented using bold or blackboard bold typeface.
Special sets of numbers include
- P or ℙ, denoting the set of all primes: P =.
- N or, denoting the set of all natural numbers: N = .
- Z or, denoting the set of all integers : Z =.
- Q or ℚ, denoting the set of all rational numbers : Q =. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can be expressed as the fraction a/1.
- R or, denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers.
- C or ℂ, denoting the set of all complex numbers: C =. For example, 1 + 2i ∈ C.
- H or ℍ, denoting the set of all quaternions: H =. For example, 1 + i + 2j − k ∈ H.
Positive and negative sets are sometimes denoted by superscript plus and minus signs, respectively. For example, ℚ^{+} represents the set of positive rational numbers.
Basic operations
There are several fundamental operations for constructing new sets from given sets.Unions
Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things that are members of either A or B.Examples:
- ∪ =
- if and only if
Intersections
Examples:
- if and only if
Complements
In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, is called the absolute complement or simply complement of A, and is denoted by A′.
- If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E = E′ = O.
- for.
- and
- and
- .
- if then
For example, the symmetric difference of and is the set. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
Cartesian product
A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs such that a is a member of A and b is a member of B.Examples:
- | A × B | = | B × A | = | A | × | B |.
Applications
One of the main applications of naive set theory is constructing relations. A relation from a domain A to a codomain B is a subset of the Cartesian product A × B. For example, considering the set S = of shapes in the game of the same name, the relation "beats" from S to S is the set B = ; thus x beats y in the game if the pair is a member of B. Another example is the set F of all pairs, where x is real. This relation is a subset of R × R, because the set of all squares is subset of the set of all real numbers. Since for every x in R', one, and only one, pair is found in F, it is called a function. In functional notation, this relation can be written as F = x^{2}.
Axiomatic set theory
Although initially naive set theory, which defines a set merely as any well-defined collection, was well accepted, it soon ran into several obstacles. It was found that this definition spawned :Category:Paradoxes of naive set theory|several paradoxes, most notably:- Russell's paradoxIt shows that the "set of all sets that do not contain themselves," i.e. the "set" does not exist.
- Cantor's paradoxIt shows that "the set of all sets" cannot exist.
For most purposes, however, naive set theory is still useful.
Principle of inclusion and exclusion
The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets, if the size of each set and the size of their intersection are known. It can be expressed symbolically asA more general form of the principle can be used to find the cardinality of any finite union of sets:
De Morgan's laws
stated two laws about sets.If A and B are any two sets then,
- ′ = A′ ∩ B′
- ′ = A′ ∪ B′