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Set

2007 Schools Wikipedia Selection. 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" \left \{ x: x\mbox{ is a set
       and }x\notin x \right \} 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 = {n^2 – 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 n^2 – 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 \in and \notin respectively. So, with respect to the sets
   defined above:

          + 4 \in A and 285 \in F (since 285 = 17² − 4); but
          + 9 \notin F and \mathrm{green} \notin B .

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 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

                             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

   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, \mathbb{Q} = \left\{
       \begin{matrix}\frac{a}{b} \end{matrix}: a,b \in \mathbb{Z}, b \neq
       0\right\} . For example, \begin{matrix} \frac{1}{4} \end{matrix}
       \in \mathbb{Q} and \begin{matrix}\frac{11}{6} \end{matrix} \in
       \mathbb{Q} . All integers are in this set since every integer a can
       be expressed as the fraction \begin{matrix} \frac{a}{1}
       \end{matrix} .
     * \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
       π, 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

   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.
   A union 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

   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.
   A intersect B

                         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

   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′.
   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} = 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′

   Retrieved from " http://en.wikipedia.org/wiki/Set"
   This reference article is mainly selected from the English Wikipedia
   with only minor checks and changes (see www.wikipedia.org for details
   of authors and sources) and is available under the GNU Free
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