Set

2007 Concise Encyclopedia. Related subjects: Mathematics

In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. Although this appears to be a simple idea, sets are one of the most important and fundamental concepts in modern mathematics. The study of the structure of possible sets, set theory, is rich and ongoing.

Having only been invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics education, being introduced from primary school in many countries. Set theory can be viewed as the foundation upon which nearly all of mathematics can be derived.

Naive set theory

At the beginning of his work Beiträge zur Begründung der transfiniten Mengenlehre, Georg Cantor, the principal creator of set theory, made the following definition of a set:

“

By a set we understand any collection M of definite, distinct objects m of our perception or of our thought (which will be called the elements of M) into a whole.

”

The objects of a set are also called its 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, for instance A, B and C. Two sets A and B are said to be equal if every member of A is also a member of B and crucially, every member of B is also a member of A; this is written A = B.

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.

Axiomatic set theory

Although initially the 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 several paradoxes, most notably:

• Russel's paradox - It shows that the "set of all sets which do not contain themselves," i.e. the "set" is not well-defined.
• Cantor's paradox - It shows that "the set of all sets" cannot exist.

The reason is that the phrase well-defined is not very well-defined. It was important to free set theory of these paradoxes since entire mathematics was being redefined in terms of set theory. In an attempt to avoid these paradoxes, set theory was axiomatized based on first-order logic, and thus the axiomatic set theory was born.

For most purposes however, the naive set theory is still useful.

Applications

Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

Description of a set

Not all sets have precise descriptions; they may be arbitrary collections, with no expressible inclusion criteria.

Some sets may be described in words:

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.

By convention, a set can be defined by explicitly listing its elements between braces (sometimes called curly brackets or curly braces):

C = {4, 2, 1, 3}

D = {red, white, blue}

Two different descriptions may define the same set. Using the above examples, A and C are identical, since they have precisely the same members. The shorthand A = C is used to express this equality.

The set described by set builder notation does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list. For example,

{6, 11} = {11, 6} = {11, 11, 6, 11}.

This is because the notation { ... } merely indicates that the set being described includes each element listed; if an element is listed more than once, or if two elements are transposed, this has no effect on the resulting set.

For sets with many elements, an abbreviated list can sometimes be used. The first one thousand positive whole numbers can be described using the symbolic shorthand:

{1, 2, 3, ..., 1000},

where the ellipsis (...) indicates the list continues in the same way. Ellipses may also be used where sets extend to infinity; the set of positive even numbers can be described : {2, 4, 6, 8, ... }.

Sets, particularly more complex ones, can use a different notation. The set F, whose members are the first twenty numbers which are four less than a square integer, can be described:

F = {n² – 4 : n is an integer; and 0 ≤ n ≤ 19}

In this description, the colon (:) means such that, and the description can be interpreted as "F is the set of numbers of the form n² – 4, such that n is a whole number in the range from 0 to 19 inclusive." Sometimes the pipe notation | is used instead of the colon.

Membership

If something is or is not an element of a particular set then this is symbolised by and respectively. So, with respect to the sets defined above:

• and (since 285 = 17² − 4); but
• and .

Cardinality

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, denoted |A|=4 and |B|=3 respectively.

A set can have zero members. Such a set is called the empty set (or the null set) and is denoted by the symbol ø. Letting A be the set of all three-sided squares, it 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 have an infinite number of members; for example, the set of natural numbers is infinite. Some infinite sets have larger cardinality than others; there are more real numbers than natural numbers (in the sense of cardinality).

Subsets

If every member of set A is also a member of 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 (A is a proper subset of B) or (B is proper superset of A). However, in some literature these symbols are read the same as and , so the more explicit symbols and are often used for proper subsets and supersets.

Example:

• 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:

• 
• 

Power set

The power set of a set S can be defined as the set of all subsets of S. This includes the subsets formed from the members of S and the empty set. If a finite set S has cardinality n then the power set of S has cardinality 2ⁿ. If S is an infinite (either countable or uncountable) set then the power set of S is always uncountable. The power set can be written as 2^S.

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. Special sets of numbers include:

• , denoting the set of all primes.
• , denoting the set of all natural numbers. That is to say, = {1, 2, 3, ...}, or sometimes = {0, 1, 2, 3, ...}.
• , denoting the set of all integers (whether positive, negative or zero). So = {..., -2, -1, 0, 1, 2, ...}.
• , denoting the set of all rational numbers (that is, the set of all proper and improper fractions). So, . For example, and . All integers are in this set since every integer a can be expressed as the fraction .
• , denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, numbers which cannot be rewritten as fractions, such as π, e, and √2).
• , denoting the set of all complex numbers.

Each of these sets of numbers has an infinite number of elements, and . The primes are used less frequently than the others outside of number theory and related fields.

Basic operations

Unions

There are 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:

• {1, 2} U {red, white} = {1, 2, red, white}
• {1, 2, green} U {red, white, green} = {1, 2, red, white, green}
• {1, 2} U {1, 2} = {1, 2}

Some basic properties of unions are:

• A U B   =   B U A
• A  ⊆  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 ∩ B, is the set of all things which are members of both A and B. If A ∩ B  =  ø, then A and B are said to be disjoint.

Examples:

• {1, 2} ∩ {red, white} = ø
• {1, 2, green} ∩ {red, white, green} = {green}
• {1, 2} ∩ {1, 2} = {1, 2}

Some basic properties of intersections:

• A ∩ B   =   B ∩ A
• A ∩ B  ⊆  A
• A ∩ A   =   A
• A ∩ ø   =   ø

Complements

Two sets can also be "subtracted". The relative complement of A in B (also called the set theoretic difference of B and A), denoted by B − A, (or 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 {1,2,3}; 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:

• {1, 2} − {red, white} = {1, 2}
• {1, 2, green} − {red, white, green} = {1, 2}
• {1, 2} − {1, 2} = 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 ∩ A′ = ø
• (A′ )′ = A
• A − A = ø
• A − B = A ∩ B′