Set (mathematics)
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Set (mathematics)
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Set (mathematics)

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 This article is about 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
Set theory.
An example of a venn diagram
[Image: 220px_Venn_A_intersect_B_svg.jpg]
In mathematics, a set is a collection of distinct objects,
considered as an object in its own right. For example, the
numbers 2, 4, and 6 are distinct objects when considered
separately, but when they are considered collectively they form
a single set of size three, written {2,4,6}. Sets are one of the
most fundamental concepts in mathematics. Developed at the
end of the 19th century, set theory is now a ubiquitous part of
mathematics, and can be used as a foundation from which
nearly all of mathematics can be derived. In mathematics
education, elementary topics such as Venn diagrams are taught
at a young age, while more advanced concepts are taught as part
of a university degree. The term itself was coined by Bolzano in
his work The Paradoxes of the Infinite.
Definition
A set is a well defined collection of distinct objects. The objects
that make up a set (also known as the elements or members of a
set) can be anything: numbers, people, letters of the alphabet,
other sets, and so on. Georg Cantor, the founder of set theory,
gave the following definition of a set at the beginning of his
Beiträge zur Begründung der transfiniten Mengenlehre:[1]

A set is a gathering together into a whole of definite,
distinct objects of our perception [Anschauung] or of our
thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and
B are equal if and only if they have precisely the same elements.
[2]
As discussed below, the definition given above turned out to be
inadequate for formal mathematics; instead, the notion of a "set"
is taken as an undefined primitive in axiomatic set theory, and
its properties are defined by the Zermelo–Fraenkel axioms. The
most basic properties are that a set "has" elements, and that two
sets are equal (one and the same) if and only if every element of
one is an element of the other.
Describing sets
There are two ways of describing, or specifying the members of,
a set. One way is by intensional definition, 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.
The second way is by extension – that is, listing each member of
the set. An extensional definition is denoted by enclosing the
list of members in curly brackets:
C = {4, 2, 1, 3}
D = {blue, white, red}.
Every element of a set must be unique; no two members may be
identical. (A multiset is a generalized concept of a set that
relaxes this criterion.) All set operations preserve this property.
The order in which the elements of a set or multiset are listed is
irrelevant (unlike for a sequence or tuple). Combining these two
ideas into an example
{6, 11} = {11, 6} = {11, 6, 6, 11}
because the extensional specification means merely that each of
the elements listed is a member of the set.
For sets with many elements, the enumeration of members can
be abbreviated. For instance, the set of the first thousand
positive integers may be specified extensionally as:
{1, 2, 3, ..., 1000},
where the ellipsis ("...") indicates that 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 notation with braces may also be used in an intensional
specification of a set. In this usage, the braces have the meaning
"the set of all ...". So, E = {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 = {n2 − 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 n2 − 4, such that n is a whole number in the range from
0 to 19 inclusive." Sometimes the vertical bar ("|") is used
instead of the colon.
One often has the choice of specifying a set intensionally or
extensionally. In the examples above, for instance, A = C and B
= D.
Membership
Main article: Element (mathematics)
The key relation between sets is membership – when one set is
an element of another. If a is a member of B, this is denoted a ∈
B, while if c is not a member of B then c ∉ B. For example, with
respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F =
{n2 − 4 : n is an integer; and 0 ≤ n ≤ 19} defined above,
4 ∈ A and 12 ∈ F; but
9 ∉ F and green ∉ B.

Subsets
Main article: 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 ⊆ B (also pronounced A is
contained in B). Equivalently, we can write B ⊇ A, 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 ⊊ B (A is a proper subset of B) or B ⊋ A (B
is a proper superset of A).
Note that the expressions A ⊂ B and B ⊃ A are used differently by
different authors; some authors use them to mean the same as A
⊆ B (respectively B ⊇ A), whereas other use them to mean the
same as A ⊊ B (respectively B ⊋ A).

[Image: 155px_Venn_A_subset_B_svg.jpg]
A is a subset of B
Example:
The set of all men is a proper subset of the set of all people.
{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.
A ⊆ A.
An obvious but useful identity, which can often be used to show
that two seemingly different sets are equal:
A = B if and only if A ⊆ B and B ⊆ A.
A partition of a set S is a set of nonempty subsets of S such that
every element x in S is in exactly one of these subsets.
Power sets
Main article: Power set
The power set of a set S is the set of all subsets of S, including S
itself and the empty set. For example, the power set of the set {1,
2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The
power set of a set S is usually written as P(S).
The power set of a finite set with n elements has 2n elements.
This relationship is one of the reasons for the terminology
power set[citation needed]. For example, the set {1, 2, 3}
contains three elements, and the power set shown above
contains 23 = 8 elements.
The power set of an infinite (either countable or uncountable)
set is always uncountable. Moreover, the power set of a set is
always strictly "bigger" than the original set in the sense that
there is no way to pair the elements of a set S with the elements
of its power set P(S) such that every element of S set is paired
with exactly one element of P(S), and every element of P(S) is
paired with exactly one element of S. (There is never a bijection
from S onto P(S).)
Every partition of a set S is a subset of the powerset of S.
Cardinality
Main article: Cardinality
The cardinality | S | of a set S is "the number of members of S."
For example, if B = {blue, white, red}, | B | = 3.
There is a unique set with no members and zero cardinality,
which is called the empty set (or the null set) and is denoted by
the symbol ∅ (other notations are used; see empty set). For
example, the set of all three-sided squares has zero members
and thus is the empty set. Though it may seem trivial, the empty
set, like the number zero, is important in mathematics; indeed,
the existence of this set is one of the fundamental concepts of
axiomatic set theory.
Some sets have infinite cardinality. The set N of natural
numbers, for instance, is infinite. Some infinite cardinalities are
greater than others. For instance, the set of real numbers has
greater 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 the entire plane, and indeed of any
finite-dimensional Euclidean space.
Special sets
There are some sets that 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, denoted {} or ∅. Another is the unit set
{x}, which contains exactly one element, namely x.[2] Many of
these sets are represented using blackboard bold or bold
typeface. Special sets of numbers include:
P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13,
17, ...}.
N or ℕ, denoting the set of all natural numbers: N = {1, 2,
3, . . .} (sometimes defined containing 0).
Z or ℤ, denoting the set of all integers (whether positive,
negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.
Q or ℚ, denoting the set of all rational numbers (that is, the set
of all proper and improper fractions): Q = {a/b : a, b ∈ Z, b ≠
0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this
set since every integer a can be expressed as the fraction a/1
(Z ⊊ Q).
R or ℝ, denoting the set of all real numbers. This set includes
all rational numbers, together with all irrational numbers (that
is, numbers that cannot be rewritten as fractions, such as √2, as
well as transcendental numbers such as π, e and numbers that
cannot be defined).
C or ℂ, denoting the set of all complex numbers: C = {a + bi :
a, b ∈ R}. For example, 1 + 2i ∈ C.
H or ℍ, denoting the set of all quaternions: H = {a + bi + cj +
dk : a, b, c, d ∈ R}. For example, 1 + i + 2j − k ∈ H.
Positive and negative sets are denoted by a superscript - or +,
for example: ℚ+ represents the set of positive rational numbers.
Each of the above sets of numbers has an infinite number of
elements, and each can be considered to be a proper subset of
the sets listed below it. The primes are used less frequently than
the others outside of number theory and related fields.
Basic operations
There are several fundamental operations for constructing new
sets from given sets.
Unions

[Image: 220px_Venn0111_svg.jpg]
The union of A and B, denoted A ∪ B
Main article: Union (set theory)
Two sets can be "added" together. The union of A and B,
denoted by A ∪ B, is the set of all things that are members of
either A or B.
Examples:
{1, 2} ∪ {1, 2} = {1, 2}.
{1, 2} ∪ {2, 3} = {1, 2, 3}.
Some basic properties of unions:
A ∪ B = B ∪ A.
A ∪ (B ∪ C) = (A ∪ B) ∪ C.
A ⊆ (A ∪ B).
A ∪ A = A.
A ∪ ∅ = A.
A ⊆ B if and only if A ∪ B = B.
Intersections
Main article: 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 that are members of
both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.

[Image: 220px_Venn0001_svg.jpg] The intersection of A and B, denoted A ∩ B.
Examples:
{1, 2} ∩ {1, 2} = {1, 2}.
{1, 2} ∩ {2, 3} = {2}.
Some basic properties of intersections:
A ∩ B = B ∩ A.
A ∩ (B ∩ C) = (A ∩ B) ∩ C.
A ∩ B ⊆ A.
A ∩ A = A.
A ∩ ∅ = ∅.
A ⊆ B if and only if A ∩ B = A.
Complements

[Image: 220px_Venn0100_svg.jpg]
The relative complement of B in A

[Image: 220px_Venn1010_svg.jpg]
The complement of A in U

[Image: 220px_Venn0110_svg.jpg]
The symmetric difference of A and B
Main article: Complement (set theory)
Two sets can also be "subtracted". The relative complement of B
in A (also called the set-theoretic difference of A and B),
denoted by A \ B (or A − B), is the set of all elements that are
members of A but not members of B. Note that it is valid to
"subtract" members of a set that are not in the set, such as
removing the element green from the set {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} \ {1, 2} = ∅.
{1, 2, 3, 4} \ {1, 3} = {2, 4}.
If U is the set of integers, E is the set of even integers, and
O is the set of odd integers, then U \ E = E′ = O.
Some basic properties of complements:
A \ B ≠ B \ A for A ≠ B.
A ∪ A′ = U.
A ∩ A′ = ∅.
(A′)′ = A.
A \ A = ∅.
U′ = ∅ and ∅′ = U.
A \ B = A ∩ B′.
An extension of the complement is the symmetric difference,
defined for sets A, B as
For example, the symmetric difference of {7,8,9,10} and
{9,10,11,12} is the set {7,8,11,12}.
Cartesian product
Main article: Cartesian product
A new set can be constructed by associating every element of
one set with every element of another set. The Cartesian
product of two sets A and B, denoted by A × B is the set of all
ordered pairs (a, b) such that a is a member of A and b is a
member of B.
Examples:
{1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2,
white)}.
{1, 2} × {red, white, green} = {(1, red), (1, white), (1,
green), (2, red), (2, white), (2, green) }.
{1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
Some basic properties of cartesian products:
A × ∅ = ∅.
A × (B ∪ C) = (A × B) ∪ (A × C).
(A ∪ B) × C = (A × C) ∪ (B × C).
Let A and B be finite sets. Then
| A × B | = | B × A | = | 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.
One of the main applications of naive set theory is constructing
relations. A relation from a domain A to a codomain B is a
subset of the Cartesian product A × B. Given this concept, we
are quick to see that the set F of all ordered pairs (x, x2), where
x is real, is quite familiar. It has a domain set R and a codomain
set that is also R, because the set of all squares is subset of the
set of all reals. If placed in functional notation, this relation
becomes f(x) = x2. The reason these two are equivalent is for
any given value, y that the function is defined for, its
corresponding ordered pair, (y, y2) is a member of the set F.
Axiomatic set theory
Main article: Axiomatic set theory
Although initially 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:
Russell's paradox—It shows that the "set of all sets that do not
contain themselves," i.e. the "set" { x : x is a set and x ∉ x }
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
because nearly all of 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 axiomatic
set theory was born.
For most purposes however, naive set theory is still useful.
Principle of inclusion and exclusion
Main article: Inclusion-exclusion principle
This principle gives us the cardinality of the union of sets.

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03-25-2014 07:22 PM
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