460 set theory

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Every infinite decimal expansion can be thought of

as an infinite series. For example,

, and the series converges to the

value 1/3.

See also

ALTERNATING SERIES

;

ARITHMETIC SERIES

;

EXPONENTIAL SERIES

;

GEOMETRIC SERIES

; G

REGORY

SERIES

;

HARMONIC SERIES

; M

ACLAURIN SERIES

;

POWER

SERIES

;

SUMS OF POWERS

; T

AYLOR SERIES

; Z

ENO

’

S PARA

-

DOXES

;

ZETA FUNCTION

.

set theory Loosely speaking, a set is any collection of

objects or numbers specified in a well-defined manner.

Each item in the set is called an element, or a member,

of the set. For example, “dog” is an element of the set

of mammals. If an entity ais an element of a set S, we

write a∈S. If adoes not belong to S, we write a∉S.

Sets are typically specified either by listing the ele-

ments of the set between a set of braces “{ }”, or listing a

few elements of the set to indicate a pattern. For exam-

ple {a,e,i,o,u} is the set consisting of the five vowels of

the alphabet, and {3,6,9,12,…} is the set of all multiples

of 3. It may also be possible to define a set as consisting

of elements from some universal collection that satisfy a

certain property. For example, {x∈R|x> 5} denotes the

set of all real numbers that are greater than 5. (Some

mathematicians prefer to use a colon “:” instead of a

vertical bar in this notation.)

The order in which the elements of a set are listed

is immaterial. For example, {A,6,*} and {*,6,A} are the

same set. Also, elements of a set are listed without repe-

tition. For instance, the set {a, a, a, a, a} is the set with

a single element a. The

EMPTY SET

is the set that con-

tains no elements.

Two sets are deemed equal if they possess precisely

the same elements. For example, the sets {2,4,6,8,…}

and {n|nis a counting number divisible by 2} are equal

sets. A set Ais said to be a subset of a set Bif every ele-

ment of Ais also a member of B. We write A⊂Bif we

are certain that the two sets are not equal, and ABif

equality of the sets is possible. For example, the set of all

multiples of 4 is a subset of the set of all multiples of 2.

Although the intuitive notion of a set as a collection

of objects is as ancient as the human race, the idea of a set

as a formal mathematical concept was not proposed until

the 19th century. In his development of B

OOLEAN ALGE

-

BRA

, British mathematician G

EORGE

B

OOLE

(1815–64)

introduced the notion of set as a fundamental tool for

the study of

FORMAL LOGIC

. German mathematician

G

EORG

C

ANTOR

(1845–1918), in his attempts to under-

stand the foundation of all of mathematics, came to

regard sets as even more basic and fundamental than the

notion of “number.” Cantor properly formalized a the-

ory of set manipulations and introduced the striking

notion of

CARDINALITY

. His work led him to profound

insights into the nature of finite and infinite sets, leading

him to extend the concept of number to include more

than one type of

INFINITY

.

Set Operations

There are a number of basic set manipulations, each of

which can be depicted with a V

ENN DIAGRAM

.

Set Intersection: The intersection of two sets Aand B,

denoted A∩B, is the set of elements common to

both Aand B. For example, if A= {0,2,4,6,8,10,12}

and B= {0,3,6,9,12,15}, then A∩B= {0, 6, 12}. Two

sets with no elements in common are called disjoint.

The intersection of two disjoint sets is the empty set.

Set Union: The union of two sets Aand B, denoted

A∪B, is the set of all elements that appear either in

Aor in B, or in both. For instance, in the previous

example we have A∪B= {0,2,3,4,6,8,10,12,15}.

If two sets Aand Beach contain a finite number

of elements, then the number of elements in A∪B

equals the number of elements in Aplus the number

of elements in Bminus the number of elements in A

∩B. (This subtraction counteracts the double count-

ing of the elements that belong to both sets.) This

formula is one instance of the general

INCLUSION

-

EXCLUSION PRINCIPLE

.

The notions of set intersection and set union

can be extended to considerations including more

than two sets.

Set Complement: If a set Ais a subset of a set B, then

the complement of Ain B, also called the set differ-

ence and denoted B– A, is the set of all elements of B

that do not belong to A. For example, if A= {1,3,5}

and B= {1,2,3,4,5,6}, then B– A= {2,4,6}. D

E

M

OR

-

GAN

’

SLAWS

explain how set complement interacts

with intersections and unions of sets.

Philosophical Difficulty

In 1902 British mathematician and philosopher

B

ERTRAND

A

RTHUR

W

ILLIAM

R

USSELL

(1872–1970)

⊇