Set

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This article is about sets in mathematics. For other meanings, see Set (disambiguation).


Sets are one of the most important and fundamental concepts in modern mathematics. Basic 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 elementary school . 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 axiomatic treatment of sets see Axiomatic set theory.

Introduction

Informally, a set is just 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, etc. Sets will usually be denoted by 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.

Ways of describing a set

A set may be described by words, for example:

A = the first three natural numbers greater than zero
B = the colors red, white, blue, and green

Another way to describe a set is to list its elements between curly braces, for example:

C = {1, 2, 3}
D = {red, white, blue, green}

Even though two sets may be described differently, they still may be identical as sets. For example, for the sets described above, A = C and B = D, since they have precisely the same members.

It makes no difference in what order the elements are listed, or whether there are repetitions in the list. For example, the three sets: {2, 4}, {4, 2}, and {2, 2, 4, 2} are identical, since again, they all have the same members.

The number of members in a set

Each of the sets described above, have a definite number of members, for example the set A has three members, while the set B has four members.

A set can also have zero members. Such a set is called the empty set (or the null set) and is denoted by the symbol ∅. For example, as of 2004, the set A of all people living on the moon, has zero members, and thus, A = ∅. Like the number zero, seemingly trivial, the empty set turns out to be quite important in mathematics.

For more information on the empty set see Empty set.

A set can also have an infinite number of members. For example the set of natural numbers is infinite.

For more information on infinity and the size of sets see Cardinal number.

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 AB. If A is subset of B, and A is not equal to B, then A is called a proper subset of B, written A ⊂ B.

A is a subset of B
A is a subset of B

Examples:

  • The set of all men is a proper subset of the set of all people.
  • The set of all natural numbers is a proper subset of all integers.
  • {1, 3} ⊂ {1, 2, 3, 4}
  • {1, 2, 3, 4} ⊆ {1, 2, 3, 4}

The empty set is a subset of every set and every set is a subset of itself:

  • ∅ ⊆ A
  • AA

For more information about subsets, see Subset.

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 ∪ B, is the set of all things which are members of either A or B.

A union B
The union of A and B

Examples:

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

Some basic properties of unions:

  • A ∪ B   =   B ∪ A
  • A  ⊆  A ∪ B
  • A ∪ A   =  A
  • A ∪ ∅   =  A

For more information about unions of sets, see Union (set theory).

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  =  ∅, then A and B are said to be disjoint.

The intersection of A and B

Examples:

  • {1, 2} ∩ {red, white} = ∅
  • {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 ∩ ∅   =   ∅

For more information about intersections of sets, see Intersection (set theory).

Complements

Two sets can also be "subtracted". The relative complement of A in B, denoted by B − A, is the set of all things which are members of B, but not members of A.

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′.

B minus A
The relative complement
of A in B
A complement
The complement of A in U

Examples:

  • {1, 2} − {red, white} = {1, 2}
  • {1, 2, green} − {red, white, green} = {1, 2}
  • {1, 2} − {1, 2} = ∅
  • If U is the set of integers, then the complement of the even integers is the odd integers

Some basic properties of complements:

  • A ∪ A′ = U
  • A ∩ A′ = ∅
  • (A′)′ = A
  • A − B = A ∩&nbspB′

For more information about complements of sets, see Complement (set theory).

Further reading

For more information on the basics properties of sets, subsets, intersections, unions and complements, see The algebra of sets. For a more general development of these ideas and others in set theory, see Naive set theory.

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