SET

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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.
''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.''

Contents

Definition

Describing sets

Membership

Cardinality

Subsets

Power set

Special sets

Basic operations

Unions

Intersections

Complements

Applications

Axiomatic set theory

See also

Notes and references

Further reading

External links

Definition

At the beginning of his work ''Beiträge zur Begründung der transfiniten Mengenlehre'', Georg Cantor, the principal creator of set theory, gave the following definition of a set:^[1]
The elements of a set, also called its ''members'', 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''. The assertion that sets ''A'' and ''B'' are 'equal' means that they have precisely the same members, that every member of ''A'' is also a member of ''B'' and every member of ''B'' is also a member of ''A''.
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

In general, there are two ways of describing any set ''S'', which is to say specifying exactly what are the members of ''S'' and what are not. One way is to define ''S'' intensionally, using a rule or semantic description:
:''A'' is the set whose members are the first four positive integers.
:''B'' is the set of colors of the French flag.
But not every set has an intensional description that is both precise and convenient. For some sets the most concise way that membership can be expressed is by explicit enumeration. In fact, this approach of enumerating its members is the second way of describing a set. Notationally, such an extensional definition is enclosed in braces (sometimes called curly brackets or curly braces):
:''C'' = {4, 2, 1, 3}
:''D'' = {blue, white, red}
The order in which the elements of a set are listed in an extensional definition is irrelevant, as are any repetitions in the list. For example,
:{6, 11} = {11, 6} = {11, 11, 6, 11}
because the extensional specification means merely that each of the elements listed is a member of the set being described.
For sets with many elements, the enumeration can be abbreviated. For instance, the set of the first thousand positive whole numbers can be specified as:
:{1, 2, 3, ..., 1000},
where the ellipsis ('...') indicates the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members. Thus the set of positive even numbers can be written as {2, 4, 6, 8, ... }.
The brace notation can also be used with an intensional specification of a set. In this usage, the braces have the meaning "the set of all..." So {playing-card suits} is the set whose four members are ♠, ♦, ♥, and ♣. A more general form of this is set-builder notation, through which, for instance, the set ''F'' of the twenty smallest integers that are four less than perfect squares can be denoted:
:''F'' = {$n^2$ – 4 ':' ''n'' is an integer; and 0 ≤ ''n'' ≤ 19}
In this notation, the colon (':') means ''such that'', and the description can be interpreted as "''F'' is the set of all numbers of the form $n^2$ – 4, where ''n'' is a whole number in the range from 0 to 19 inclusive." Sometimes the pipe ('|') is used instead of the colon.
One often has the choice of specifying a set intensionally or extensionally. In the examples above, for instance, note that ''A'' = ''C'' and ''B'' = ''D''.

Membership

Main articles: Element (mathematics)

If something is or is not an element of a particular set then this is symbolised by $in$ and $$
otin respectively. So, with respect to the sets defined above:
:
★ $4\; in\; A$ and $285\; in\; F$ (since 285 = 17² − 4); but
:
★ $9$
otin F and $mathrm\{green\}$
otin B.

Cardinality

Main articles: Cardinality

The cardinality |''S''| of a set ''S'' is "the number of members of ''S''." For example, since the French flag has three colors, |B| = 3.
A set can have zero members. Such a set is called an ''empty set'' (or a ''null set'') and is denoted by the symbol ø. For example, the set ''A'' of all three-sided squares has zero members (|''A''| = 0), and thus ''A'' = ø. Though, like the number zero, it may seem trivial, the empty set is quite important in mathematics.
Some sets have infinite cardinality. The set 'N' of natural numbers, for instance, is infinite. And some infinite cardinalities are larger than others. For instance, the set of real numbers has larger cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of an entire plane, and indeed of an ''n''-dimensional Euclidean space for any finite value of ''n''.

Subsets

Main articles: Subset

If every member of set ''A'' is also a member of set ''B'', then ''A'' is said to be a ''subset'' of ''B'', written $A\; subseteq\; B$ (also pronounced ''A is contained in B''). Equivalently, we can write $B\; supseteq\; A$, read as ''B is a superset of A'', ''B includes A'', or ''B contains A''. The relationship between sets established by $subseteq$ 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 $A\; subset\; B$ (''A is a proper subset of B'') or $B\; supset\; A$ (''B is proper superset of A''). However, in some literature these symbols are read the same as $subseteq$ and $supseteq$, so the more explicit symbols $subsetneq$ and $supsetneq$ are often used for proper subsets and supersets.
''A'' is a 'subset' of ''B''
Example:
:
★ The set of all men is a proper subset of the set of all people.
:
★ $\{1,3\}\; subset\; \{1,2,3,4\}$
:
★ $\{1,\; 2,\; 3,\; 4\}\; subseteq\; \{1,2,3,4\}$
The empty set is a subset of every set and every set is a subset of itself:
:
★ $emptyset\; subseteq\; A$
:
★ $A\; subseteq\; A$

Power set

Main articles: 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 2^''n''. 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:

★ $mathbb\{P\}$, denoting the set of all primes.

★ $mathbb\{N\}$, denoting the set of all natural numbers. That is to say, $mathbb\{N\}$ = {1, 2, 3, ...}, or sometimes $mathbb\{N\}$ = {0, 1, 2, 3, ...}.

★ $mathbb\{Z\}$, denoting the set of all integers (whether positive, negative or zero). So $mathbb\{Z\}$ = {..., -2, -1, 0, 1, 2, ...}.

★ $mathbb\{Q\}$, denoting the set of all rational numbers (that is, the set of all proper and improper fractions). So,
eq 0
ight}. For example, and . All integers are in this set since every integer ''a'' can be expressed as the fraction .

★ $mathbb\{R\}$, 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 $pi,$ $e,$ and √2).

★ $mathbb\{C\}$, denoting the set of all complex numbers.
Each of these sets of numbers has an infinite number of elements, and $mathbb\{P\}\; subset\; mathbb\{N\}\; subset\; mathbb\{Z\}\; subset\; mathbb\{Q\}\; subset\; mathbb\{R\}\; subset\; mathbb\{C\}$. The primes are used less frequently than the others outside of number theory and related fields.

Basic operations

Unions

Main articles: Union (set theory)

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''.
The 'union' of ''A'' and ''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

Main articles: Intersection (set theory)

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''.
The 'intersection' of ''A'' and ''B''
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

Main articles: Complement (set theory)

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''′.
The 'relative complement'
of ''A'' in ''B''
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} = 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′''

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.

Axiomatic set theory

Main articles: 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 , most notably:

★ Russell's paradox - It shows that the "set of all sets which ''do not contain themselves''," i.e. the "set" $left\; \{\; x:\; xmbox\{\; is\; a\; set\; and\; \}x$
otin x
ight } does not exist.

★ 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.

See also

★ Alternative set theory ★ Axiomatic set theory ★ Class (set theory) ★ Dense set ★ Family (mathematics) ★ Fuzzy set ★ Mathematical structure

★ Multiset ★ Naive set theory ★ Rough set ★ Russell's paradox ★ Scientific classification ★ Taxonomy ★ Tuple

Notes and references

1. Allenby, p. 1.

Further reading

★ Halmos, Paul R., ''Naive Set Theory'', Princeton, N.J.: Van Nostrand (1960) ISBN 0-387-90092-6

★ Stoll, Robert R., ''Set Theory and Logic'', Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4

★ Allenby, R.B.J.T, ''Rings, Fields and Groups'', Leeds, England: Butterworth Heinemann (1991) ISBN 0-340-54440-6

External links

★ C2 Wiki - Examples of set operations using English operators.