set

set

\PMlinkescapephrase

basic set

1

Introduction

A

set

is a

collection

, , or conglomerate

1

1

However, not every collection has to be a set (in fact, all collections

can’t

be sets: there is no set of all sets or of all

ordinals

for example). See

proper class

for more details.

.

Sets can be of “” objects or mathematical objects, but the sets themselves are purely conceptual. This is an important point to note: the set of all cows (for example) does not physically exist, even though the cows do. The set is a “gathering” of the cows into one conceptual that is not part of physical reality. This makes it easy to see why we can have sets with an

infinite

number of

elements

; even though we may not be able to point out infinitely many objects in the real world, we can

construct

conceptual sets which an infinite number of elements (see the examples below).

The symbol

∈

denotes set

membership

. For example,

s

∈

S

would be read “

s

is an element of

S

”.

We write

A

⊂

B

if for all

x

⁢

ϵ

⁢

A

we have

x

⁢

ϵ

⁢

B

and we then say

B

contains

A

.
We sometimes write
“

S

contains

s

” when

S

contains the
set whose only element is

s

.

Mathematics is thus built upon sets of purely conceptual, or mathematical, objects. Sets are usually denoted by upper-case roman letters (such as

S

). Sets can be defined by listing the members, as in

S

:=

{

a

,

b

,

c

,

d

}

.

Or, a set can be defined from a

predicate

(called “set builder notation”). This type of statement defining a set is of the form

S

:=

{

x

∈

X

:

P

⁢

(

x

)

}

,

where

S

is the symbol denoting the set,

x

is the

variable

we are introducing to

represent

a

generic

element of the set (note that, by the so called

axiom of comprehension

(or

axiom of subsets

),

x

must be a member of some set which has already been defined. This is necessary in order to avoid Russell’s

paradox

2

2

One needs to be careful when defining a set by a predicate only, since (for example) “

x

is not in

x

” is a perfectly good predicate. Either one needs to restrict the kind of predicate, or, more commonly, one needs to

define only subsets

by predicates. So while one cannot do

{

x

:

x

∉

x

}

, if one already has a set

U

, one can do

{

x

:

x

∈

U

,

x

∉

x

}

.

.) and

P

⁢

(

x

)

is some property which must be true for any element

x

of the set (that is,

x

∈

S

is

equivalent

to

x

∈

X

and

P

⁢

(

x

)

holds.) Sometimes, we write a set definition as

S

:=

{

f

⁢

(

x

)

∈

X

:

P

⁢

(

x

)

}

, where

f

⁢

(

x

)

is a

transformation

of that variable. In this case, we can simply replace the set

X

by

Y

, where

Y

:=

{

y

∈

X

:

∃

x

∈

X

,

y

=

f

⁢

(

x

)

}

in order to define the set as above.

Sets are, in fact, completely specified by their elements. If two sets have the same elements, they are equal. This is called the

axiom of extensionality

, and it is one of the most important characteristics of sets that distinguishes them from predicates or properties.

Some examples of sets are:

•

The standard number sets

ℕ

,

ℤ

,

ℚ

,

ℝ

and

ℂ

.

•

The set of all

even integers

:

{

x

∈

ℤ

:

2

∣

x

}

.

•

The set of all prime numbers (sometimes denoted

ℙ

):

{

p

∈

ℕ

:

p

>

1

,

∀

x

∈

ℕ

x

∣

p

⇒

x

∈

{

1

,

p

}

}

, where

⇒

denotes implies and

∣

denotes divides.

•

The set of all real functions of one real parameter (sometimes denoted by

ℝ

ℝ

):

{

f

⁢

(

x

)

∈

ℝ

:

x

∈

ℝ

}

or, more formally,

{

f

⊂

ℝ

2

:

[

∀

x

∈

ℝ

,

∃

y

∈

ℝ

,

(

x

,

y

)

∈

f

]

∧

[

(

x

,

y

)

,

(

x

,

y

′

)

∈

f

⇒

y

=

y

′

]

}

.

•

The unit circle

𝕊

1

:

{

z

∈

ℂ

:

|

z

|

=

1

}

, where

|

z

|

is the modulus of

z

.

The most basic set is the

empty set

(denoted

∅

,

∅

or

{

}

).

The astute reader may have noticed that all of our examples of sets utilize sets, which does not suffice for

rigorous

definition. We can be more rigorous if we

postulate

only the empty set, and define a set in general as anything which one can construct from the empty set and the ZFC axioms. The non-negative integers, for

instance

, are defined by

0

:=

∅

and the

successor

of

x

,

s

⁢

(

x

)

=

x

∪

{

x

}

.

A non-negative integer is thus the set of all its

predecessors

(for example, we have

3

=

{

0

,

1

,

2

}

=

{

∅

,

{

∅

}

,

{

∅

,

{

∅

}

}

}

)

3

3

Note however that the

existence

of the

set

of non-negative integers needs an additional axiom beside those which are required to define its members: the

axiom of infinity

.

.

All objects in modern mathematics are constructed via sets. An important point to be made about this is that the construction of the object is less important than the way it will

behave

. As an example, we usually define an

ordered pair

(

x

,

y

)

as the set

{

x

,

{

x

,

y

}

}

: what matters here is that, for two ordered pairs

(

x

,

y

)

and

(

x

′

,

y

′

)

, we have

(

x

,

y

)

=

(

x

′

,

y

′

)

if and only if

x

=

x

′

and

y

=

y

′

, and this is true with the given definition, as one can easily see. We could, however, also have taken

{

x

,

{

x

,

{

y

}

}

}

as the definition of

(

x

,

y

)

, in which case the needed property also holds and we have a valid definition (we chose the first only because it is simpler).

2

Set Notions

An important set notion is

cardinality

. Cardinality is roughly the same as the intuitive notion of “size” or number of elements. While this intuitive definition works well for

finite sets

, intuition breaks down for sets with an infinite number of elements. The cardinality of a set

S

is denoted

|

S

|

(sometimes

#

⁢

S

or

card

⁡

S

) and we say that sets

A

and

B

have the same cardinality if and only if there is a

bijection

from one to the other. For more detail, see the cardinality entry.

Another important set

concept

is that of

subsets

. A subset

B

of a set

A

is any set which contains only elements that appear in

A

. Subsets are denoted with the

⊆

symbol, i.e.

B

⊆

A

(in which case

A

is called a

superset

of B). Also useful is the notion of a

proper subset

, denoted

B

⊊

A

(or sometimes,

B

⊂

A

)

4

4

Beware — some authors use

⊂

to mean proper subset, while most use it to mean subset with

equality

(the same as

⊆

), which can make the

B

⊂

A

notation ambiguous.

, which adds the

restriction

that

B

must also not be equal to

A

. The set of all subsets of a set

S

is called

the

power set

of

S

, denoted

𝒫

⁢

(

S

)

(the existence of this set is also axiomatic: it is guaranteed by the axiom of the power set). Note that

B

does not need to have a lower cardinality than

A

to be a proper subset, i.e.,

{

1

,

2

,

3

,

…

}

is a proper subset of

{

0

,

1

,

2

,

3

,

…

}

, but both have the same cardinality,

ℵ

0

(In fact, a set is infinite if and only if it has the same cardinality as some proper subset).

3

Set Operations

There are a number of standard (common)

operations

which are used to manipulate sets, producing new sets from

combinations

of existing sets (sometimes with entirely different types of elements). These standard operations are:

•

union (

http://planetmath.org/Union

)

•

intersection

•

set difference

•

symmetric set difference

•

complement

•

Cartesian product

•

power set (

http://planetmath.org/PowerSet

)

Title

set

Canonical name

Set

Date of creation

2013-03-22 12:15:09

Last modified on

2013-03-22 12:15:09

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

24

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 03E99

Related topic

ZermeloFraenkelAxioms

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Mapping

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SetTheory

Related topic

Class

Related topic

DeMorgansLaws

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NotationInSetTheory

Related topic

CardinalityOfAFiniteSetIsUnique

Related topic

OneToOneFunctionFromOntoFunction

Related topic

Collection

Related topic

FunctorCategory2

Related topic

Algebras2

Related topic

SetMembership

Defines

contains

Defines

subset

Defines

proper subset

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