Encyclopedia
In
mathematics, a
set can be thought of as any collection of distinct things considered as a whole. Though a simple idea, it is nevertheless one of the most important and fundamental concepts in modern
mathematics, and the study of the structure of possible sets, set theory, is quite rich.
Set theory, having only been invented at the end of the
19th century, is now a ubiquitous part of mathematics education, being introduced as early as
primary school. It is the language in which modern mathematics is described. Set theory can be viewed as the foundation upon which nearly all of mathematics can be built and the source from which nearly all mathematics can be derived.
This article gives a brief and basic introduction to what mathematicians call "intuitive" or "naive" set theory; for a more detailed account see naive set theory. For a rigorous modern axiomatic treatment of sets see axiomatic set theory.
Definition
A
set is a collection of objects considered as a whole. The objects of a set are called
elements or
members. The elements of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters,
A,
B,
C, etc. Two sets
A and
B are said to be equal, written
A =
B, if they have the same members.
A set, unlike a multiset, cannot contain two or more identical elements. All set operations preserve the property that each element in the set is unique. Similarly, the order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple.
Describing sets
Not all sets have precise descriptions of any sort; they may simply be arbitrary collections, with no expressible "rule" saying what elements are in or out.
Some sets may be described in words, for example:
- A is the set whose members are the first four positive whole numbers.
- B is the set whose members are the colors of the French flag
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By convention, a set can also be defined by explicitly listing its elements between
braces , for example:
- C =
- D =
Two different descriptions may define the same set. For example, for the sets defined above,
A and
C are identical, since they have precisely the same members. The shorthand
A =
C is used to express this equality. Similarly, for the sets defined above,
B =
D.
Set identity does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list. For example, = = .
For sets with many elements, an abbreviated list is sometimes used. For example, the first one thousand positive whole numbers can be described using the symbolic shorthand:
- ,
where the ellipsis indicates that the list continues in the obvious way.
Similarly the set of even numbers can be described by the notation:
- .
More complicated sets are sometimes described by a different notation. For example the set
F, whose members are the first twenty numbers which are four less than a square integer, can be described using the following:
- F =
In this description, the colon means "such that", and the mathematician interprets this description as "
F is the set of numbers of the form – 4, such that
n is a whole number in the range from 0 to 19 inclusive."
For more information on describing sets see Set-builder notation.
Set membership
If something is or is not an element of a particular set then this is symbolised by and respectively. So, for example, with respect to the sets defined above:
Cardinality of a set
Each of the sets described above has a definite number of members; for example, the set
A has four members, while the set
B has three members.
A set can also have zero members. Such a set is called the
empty set and is denoted by the symbol ø. For example, the set
A of all three-sided squares has zero members, and thus
A = ø. Like the number zero, though seemingly trivial, the empty set turns out to be quite important in mathematics.
A set can also have an
infinite number of members; for example, the set of natural numbers is infinite.
Subsets
If every member of the set
A is also a member of the set
B, then
A is said to be a
subset of
B, written , also pronounced
A is contained in B. Equivalently, we can write , 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.
If
A is a subset of but not equal to
B, then
A is called a
proper subset of
B, written or . However, in some literature these symbols are read the same as and , so it's often preferred to use the more explicit symbols and for proper subsets and supersets.
Examples:
- The set of all men is a proper subset of the set of all people.
The empty set is a subset of every set and every set is a subset of itself:
Special sets
There are some sets which 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. Many of these sets are represented using
Blackboard bold typeface. Some special sets of numbers include:
- denotes the set of all primes.
- denotes the set of all natural numbers. That is to say, = , or sometimes = .
- denotes the set of all integers . So = .
- denotes the set of all rational numbers . So, = . For example, and . All integers are in this set since every integer a can be expressed as the fraction .
- is the set of all real numbers. This set includes all rational numbers, together with all irrational numbers .
- is the set of all complex number
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Each of these sets of numbers has infinite cardinality, and moreover , although the primes are generally used less than the others outside of number theory and related fields.
Unions
There are several ways to construct new sets from existing ones.
Two sets can be "added" together. The
union of
A and
B, denoted by
A U
B, is the set of all things which are members of either
A or
B.
Examples:
Some basic properties of unions:
- A U B = B U A
- A is a subset of A U B
- A U A = A
- A U ø = A
Intersections
A new set can also be constructed by determining which members two sets have "in common". The
intersection of
A and
B, denoted by
A n
B, is the set of all things which are members of both
A and
B. If
A n
B = ø, then
A and
B are said to be
disjoint.
Examples:
Some basic properties of intersections:
- A n B = B n A
- A n B is a subset of A
- A n A = A
- A n ø = ø
Complements
Two sets can also be "subtracted". The
relative complement of
A in
B , denoted by
B −
A, is the set of all elements which are members of
B, but not members of
A. Note that it is valid to "subtract" members of a set that are not in the set, such as removing
green from ; doing so has no effect.
In certain settings all sets under discussion are considered to be subsets of a given
universal set U. In such cases,
U −
A, is called the
absolute complement or simply
complement of
A, and is denoted by
A′.
Examples:
- − =
- − =
- − = ø
- If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then the complement of E in U is O, or equivalently, E′ = O.
Some basic properties of complements:
- A U A′ = U
- A n A′ = ø
- ′ = A
- A − A = ø
- A − B = A n B′
Further reading
For more information on the basic properties of sets, subsets, intersections, unions and complements, see algebra of sets. For a more general development of these ideas and others in set theory, see naive set theory.
See also
- Alternative set theory
- Class
- Family
- Mathematical structure
- Multiset
- Russell's paradox
- Tuple
References
- Halmos, Paul R., Naive Set Theory, Princeton, N.J.: Van Nostrand ISBN 0-387-90092-6
- Stoll, Robert R., Set Theory and Logic, Mineola, N.Y.: Dover Publications ISBN 0-486-63829-4
