aarjaneiro/planetmath huggingface dataset:
indexing set

indexing set

Let

Λ

and

S

be sets such that there exists a

surjection

f

:

Λ

→

S

. Then

Λ

is an

indexing set

for

S

. Also,

S

is

indexed by

Λ

.

In such situations, the elements of

S

could be referenced by using the indexing set

Λ

, such as

f

⁢

(

λ

)

for some

λ

∈

Λ

. On the other hand, quite often, indexing sets are used without explicitly defining a surjective function. When this occurs, the elements of

S

are referenced by using

subscripts

(also called

indices

) which are elements of

Λ

, such as

s

λ

for some

λ

∈

Λ

. If, however, the surjection from

Λ

to

S

were called

s

, this notation would be quite to the

function

notation:

s

⁢

(

λ

)

=

s

λ

.

Indexing sets are quite useful for describing

sequences

, nets,

summations

,

products

, unions, and

intersections

.

Multiple indices are possible. For example, consider the set

X

=

{

x

a

⁢

a

,

x

a

⁢

b

,

x

a

⁢

c

,

x

b

⁢

b

,

x

b

⁢

c

,

x

c

⁢

c

}

. Some people would consider the indexing set for

X

to be

{

a

⁢

a

,

a

⁢

b

,

a

⁢

c

,

b

⁢

b

,

b

⁢

c

,

c

⁢

c

}

. Others would consider the indexing set to be

{

a

,

b

,

c

}

×

{

a

,

b

,

c

}

. (The double indices can be considered as

ordered pairs

.) Thus, in the case of multiple indices, it need not be the case that the underlying function

f

be a surjection. On the other hand,

f

must be a partial surjection. For example, if a set

X

is indexed by

A

×

B

, the following must hold:

1.

For every

x

∈

X

, there exist

i

∈

A

and

j

∈

B

such that

f

⁢

(

i

,

j

)

=

x

;

2.

For every

i

∈

A

, the map

f

i

:

B

→

X

defined by

f

i

⁢

(

j

)

=

f

⁢

(

i

,

j

)

is a

partial function

;

3.

For every

j

∈

B

, the map

f

j

:

A

→

X

defined by

f

j

⁢

(

i

)

=

f

⁢

(

i

,

j

)

is a partial function.

Title

indexing set

Canonical

name

IndexingSet

Date of creation

2013-03-22 16:07:51

Last modified on

2013-03-22 16:07:51

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03E99

Synonym

index set

Defines

subscript

Defines

index

Defines

indices

Defines

indexed by

Defines

double indices

Defines

multiple indices

Generated on Thu Feb 8 20:21:10 2018 by

LaTeXML
infinite

infinite

A set

S

is

infinite

if it is not finite (

http://

planetmath

.org/Finite

); that is, there is no

n

∈

ℕ

for which there is a

bijection

between

n

and

S

.

Assuming the

Axiom of Choice

(

http://planetmath.org/AxiomOfChoice

) (or the

Axiom of Countable Choice

), this definition of

infinite sets

is

equivalent

to that of Dedekind-infinite sets (

http://planetmath.org/DedekindInfinite

).

Some examples of

finite sets

:

•

The

empty set

:

{

}

.

•

{

0

,

1

}

•

{

1

,

2

,

3

,

4

,

5

}

•

{

1

,

1.5

,

e

,

π

}

Some examples of infinite sets:

•

{

1

,

2

,

3

,

4

,

…

}

.

•

The primes:

{

2

,

3

,

5

,

7

,

11

,

…

}

.

•

The

rational numbers

:

ℚ

.

•

An interval of the reals:

(

0

,

1

)

.

The first three examples are

countable

, but the last is

uncountable

.

Title

infinite

Canonical name

Infinite

Date of creation

2013-03-22 11:59:03

Last modified on

2013-03-22 11:59:03

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

18

Author

yark (2760)

Entry type

Definition

Classification

msc 03E99

Synonym

infinite set

Synonym

infinite subset

Related topic

Finite

Related topic

AlephNumbers

Generated on Thu Feb 8 20:21:13 2018 by

LaTeXML
intersection

intersection

The

intersection

of two sets

A

and

B

is the set that contains all the

elements

x

such that

x

∈

A

and

x

∈

B

. The intersection of

A

and

B

is written as

A

∩

B

. The following

Venn diagram

illustrates the intersection of two sets

A

and

B

:

Example

. If

A

=

{

1

,

2

,

3

,

4

,

5

}

and

B

=

{

1

,

3

,

5

,

7

,

9

}

then

A

∩

B

=

{

1

,

3

,

5

}

.

We can also define the intersection of an arbitrary number of sets. If

{

A

j

}

j

∈

J

is a family of sets, we define the intersection of all them, denoted

⋂

j

∈

J

A

j

, as the set consisting of those elements belonging to every set

A

j

:

⋂

j

∈

J

A

j

=

{

x

:

x

∈

A

j

⁢

for all

⁢

j

∈

J

}

.

A set

U

intersects

, or

meets

, a set

V

if

U

∩

V

is non-empty.

Some

elementary

properties

of

∩

are

•

(

idempotency

)

A

∩

A

=

A

,

•

(

commutativity

)

A

∩

B

=

B

∩

A

,

•

(

associativity

)

A

∩

(

B

∩

C

)

=

(

A

∩

B

)

∩

C

,

•

A

∩

A

∁

=

∅

, where

A

∁

is the

complement

of

A

in some fixed

universe

U

.

Remark

. What is

⋂

j

∈

J

A

j

when

J

=

∅

? In other words, what is the intersection of an empty family of sets? First note that if

I

⊆

J

, then

⋂

j

∈

J

A

j

⊆

⋂

i

∈

I

A

i

.

This leads the

conclusion

that the intersection of an empty family of sets should be as large as possible. How large should it be? In

addition

, is this intersection a set? The answer depends on what versions of

set theory

we are working in. Some theories (for example, von Neumann-Gödel-Bernays) say this is the class

V

of all sets, while others do not define this notion at all. However, if there is a

fixed set

U

in advance such that each

A

j

⊆

U

, then it is sometimes a matter of convenience to

define

the intersection of an empty family of

A

j

to be

U

.

Title

intersection

Canonical

name

Intersection

Date of creation

2013-03-22 12:14:52

Last modified on

2013-03-22 12:14:52

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

22

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E99

Synonym

intersects

Synonym

meets

Related topic

union

Related topic

Union

Related topic

FiniteIntersectionProperty

Related topic

EmptySet

Related topic

ProductOfLeftAndRightIdeal

Generated on Thu Feb 8 20:21:15 2018 by

LaTeXML
limit along a filter

limit along a filter

Definition 1

.

Let

F

be a filter on

N

and

(

x

n

)

be a

sequence

in a

metric
space

(

X

,

d

)

. We say that

L

is the

ℱ

-limit

of

(

x

n

)

if

A

⁢

(

ε

)

=

{

n

∈

ℕ

:

d

⁢

(

x

n

,

L

)

<

ε

}

∈

ℱ

for every

ε

>

0

.

The name

along

F

is used as well.

In the usual definition of limit one requires all sets

A

⁢

(

ε

)

to
be cofinite - i.e. they have to be large. In the definition of

ℱ

-limit we simply choose which sets are considered to be large
- namely the sets from the filter

ℱ

.

Remarks

This notion shouldn’t be confused with the notion of limit of a
filter (

http://planetmath.org/filter

) defined in general topology.

Let us note that the same notion is defined by some authors using
the dual notion of ideal instead of filter and, of course, all
results can be reformulated using ideals as well. For this
approach see e.g.

[

4

]

.

Examples

Limit along the Fréchet filter, which consist of

complements

of

finite sets

, is the usual limit of a sequence.

Limit of the sequence

(

x

n

)

along the

principal filter

ℱ

k

=

{

A

⊆

ℕ

;

k

∈

A

}

is

x

k

.

If we put

ℱ

=

{

ℕ

∖

A

:

d

⁢

(

A

)

=

0

}

, where

d

denotes the
asymptotic density, then it can be shown that

ℱ

is a filter. In
this case

ℱ

-convergence is known as statistical convergence.

References

1

M. A. Alekseev, L. Yu. Glebsky, and E. I. Gordon,

On
approximations of
groups, group actions and

Hopf algebras

, Journal of Mathematical Sciences

107

(2001), no. 5, 4305–4332.

2

B. Balcar and P. Štěpánek,

Teorie množin

,
Academia,
Praha, 1986 (Czech).

3

K. Hrbacek and T. Jech,

Introduction

to

set theory

,
Marcel Dekker,
New York, 1999.

4

P. Kostyrko, T. Šalát, and W. Wilczyński,

I

-convergence

, Real Anal. Exchange

26

(2000-2001), 669–686.

Title

limit along a filter

Canonical name

LimitAlongAFilter

Date of creation

2013-03-22 15:32:20

Last modified on

2013-03-22 15:32:20

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

6

Author

kompik (10588)

Entry type

Definition

Classification

msc 03E99

Classification

msc 40A05

Synonym

limit along filter

Synonym

F-limit

Related topic

Filter

Defines

limit along a filter

Generated on Thu Feb 8 20:21:18 2018 by

LaTeXML
multiplicative filter

multiplicative filter

For any ring

A

, any set

S

⊂

A

and any

element

x

∈

A

,
we use the notation

(

S

:

x

)

:=

{

a

∈

A

a

x

∈

S

}

Let

A

be a commutative ring with unity, and let

ℐ

⁢

(

A

)

be the set of all ideals of

A

.

•

A

Multiplicative Filter

of

A

is a filter

ℱ

on

ℐ

⁢

(

A

)

such that

I

,

J

∈

ℱ

⇒

I

⁢

J

∈

ℱ

.

•

A

Gabriel Filter

of

A

is a filter

ℱ

on

ℐ

⁢

(

A

)

such that

[

I

∈

ℱ

,

J

∈

ℐ

(

A

)

and

∀

x

∈

I

,

(

J

:

x

)

∈

ℱ

]

⇒

J

∈

ℱ

Note that Gabriel Filters are also Multiplicative Filters.

Title

multiplicative filter

Canonical name

MultiplicativeFilter

Date of creation

2013-03-22 16:48:22

Last modified on

2013-03-22 16:48:22

Owner

jocaps (12118)

Last modified by

jocaps (12118)

Numerical id

6

Author

jocaps (12118)

Entry type

Example

Classification

msc 03E99

Classification

msc 54A99

Defines

Gabriel Filter

Defines

Multiplicative Filter

Generated on Thu Feb 8 20:21:20 2018 by

LaTeXML
multiset

multiset

A

multiset

is a set for which repeated

elements

are considered.

Note that the standard

definition

of a set also allows repeated elements, but these are not

treated

as repeated elements. For example,

{

1

,

1

,

3

}

as a

set

is actually equal to

{

1

,

3

}

. However, as a

multiset

,

{

1

,

1

,

3

}

is not simplifiable further.

A definition that makes clear the distinction between set and multiset follows:

Multiset

.

A multiset is a pair

(

X

,

f

)

, where

X

is a set, and

f

is a

function

mapping

X

to the

cardinal numbers

greater than zero

.

X

is called the

underlying set

of the multiset, and for any

x

∈

X

,

f

⁢

(

x

)

is the

multiplicity

of

x

.

Using this definition and expressing

f

as a set of

ordered pairs

, we see that the multiset

{

1

,

3

}

has

X

=

{

1

,

3

}

and

f

=

{

(

1

,

1

)

,

(

3

,

1

)

}

. By contrast, the multiset

{

1

,

1

,

3

}

has

X

=

{

1

,

3

}

and

f

=

{

(

1

,

2

)

,

(

3

,

1

)

}

.

Generally, a multiplicity of zero is not allowed, but a few mathematicians do allow for it, such as Bogart and Stanley. It is far more common to disallow

infinite

multiplicity, which greatly complicates the definition of

operations

such as unions,

intersections

,

complements

, etc.

References

1

Kenneth P. Bogart,

Introductory

Combinatorics

. Florence, Kentucky: Cengage Learning (2000): 93

2

John L. Hickman, “A note on the

concept

of multiset”

Bulletin of the Australian Mathematical Society

22

(1980): 211 - 217

3

Richard P. Stanley,

Enumerative Combinatorics

Vol 1. Cambridge: Cambridge University Press (1997): 15

Title

multiset

Canonical

name

Multiset

Date of creation

2013-03-22 12:21:44

Last modified on

2013-03-22 12:21:44

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

13

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 03E99

Synonym

bag

Related topic

AxiomsOfMetacategoriesAndSupercategories

Related topic

ETAS

Defines

multiplicity

Generated on Thu Feb 8 20:21:22 2018 by

LaTeXML
pairwise disjoint

pairwise disjoint

Definition

Suppose

{

E

α

∣

α

∈

I

}

is an arbitrary

collection

of sets.
These sets are said to be

pairwise disjoint

if for every pair of distinct

elements

α

,

β

∈

I

,
we have

E

α

∩

E

β

=

∅

.

Remark

The synonym

mutually disjoint

is also used.

Title

pairwise disjoint

Canonical name

PairwiseDisjoint

Date of creation

2013-03-22 14:13:05

Last modified on

2013-03-22 14:13:05

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 03E99

Synonym

mutually disjoint

Generated on Thu Feb 8 20:21:38 2018 by

LaTeXML
power set

power set

Definition

If

X

is a set, then the

power set

of

X

, denoted by

𝒫

⁢

(

X

)

, is the
set whose

elements

are the subsets of

X

.

Properties

1.

If

X

is finite, then

|

𝒫

⁢

(

X

)

|

=

2

|

X

|

.

2.

The above

property

also holds when

X

is not finite.
For a set

X

, let

|

X

|

be the

cardinality

of

X

.
Then

|

𝒫

⁢

(

X

)

|

=

2

|

X

|

=

|

2

X

|

,
where

2

X

is the set of all

functions

from

X

to

{

0

,

1

}

.

3.

For an arbitrary set

X

, Cantor’s

theorem

states:
a) there is no

bijection

between

X

and

𝒫

⁢

(

X

)

, and
b) the cardinality of

𝒫

⁢

(

X

)

is greater than the cardinality of

X

.

Example

Suppose

S

=

{

a

,

b

}

. Then

𝒫

⁢

(

S

)

=

{

∅

,

{

a

}

,

{

b

}

,

S

}

.
In particular,

|

𝒫

⁢

(

S

)

|

=

2

|

S

|

=

4

.

Related definition

If

X

is a set, then the

finite power set of

X

, denoted by

ℱ

⁢

(

X

)

, is the
set whose elements are the

finite

subsets of

X

.

Remark

Due to the

canonical

correspondence between elements of

𝒫

⁢

(

X

)

and elements of

2

X

, the power set is sometimes also denoted by

2

X

.

Title

power set

Canonical name

PowerSet

Date of creation

2013-03-22 11:43:46

Last modified on

2013-03-22 11:43:46

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

23

Author

matte (1858)

Entry type

Definition

Classification

msc 03E99

Classification

msc 03E10

Classification

msc 37-01

Synonym

powerset

Related topic

PowerObject

Related topic

ProofOfGeneralAssociativity

Defines

finite power set

Defines

finite powerset

Generated on Thu Feb 8 20:21:50 2018 by

LaTeXML
proof of alternative characterization of filter

proof of alternative characterization of filter

First, suppose that

𝐅

is a filter. We shall show that, for
any two

elements

A

and

B

of

𝐅

, it is the case that

A

∩

B

∈

𝐅

if and only if

A

∈

𝐅

and

B

∈

𝐅

.

By the

definition

of filter, if

A

∈

𝐅

and

B

∈

𝐅

then

A

∩

B

∈

𝐅

. Since

A

⊇

A

∩

B

and

𝐅

is a filter,

A

∩

B

∈

𝐅

implies

A

∈

𝐅

. Likewise,

A

∩

B

∈

𝐅

implies

B

∈

𝐅

.

Next, we shall show that any

proper subset

𝐅

of the

power
set

of

X

such that

A

∩

B

∈

𝐅

if and only if

A

∈

𝐅

and

B

∈

𝐅

is a filter.

If the

empty set

were to belong to

𝐅

then for any

A

⊂

X

, we would have

A

∩

∅

=

∅

∈

𝐅

. This would imply that every subset of

X

belongs to

𝐅

, contrary to our

hypothesis

that

𝐅

is a proper
subset of the power set of

X

.

If

A

⊆

B

⊆

X

and

A

∈

𝐅

, then

A

∩

B

=

A

∈

𝐅

. By our hypothesis,

B

∈

𝐅

.

The third

defining property

of a filter — If

A

∈

𝐅

and

B

∈

𝐅

then

A

∩

B

∈

𝐅

— is part of our
hypothesis.

Title

proof of alternative characterization of filter

Canonical

name

ProofOfAlternativeCharacterizationOfFilter

Date of creation

2013-03-22 14:43:05

Last modified on

2013-03-22 14:43:05

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 03E99

Classification

msc 54A99

Generated on Thu Feb 8 20:21:52 2018 by

LaTeXML
proof of delta system lemma

proof of delta system lemma

Since there are only

ℵ

0

possible

cardinalities

for any element of

S

, there must be some

n

such that there are an

uncountable

number of elements of

S

with cardinality

n

. Let

S

*

=

{

a

∈

S

∣

|

a

|

=

n

}

for this

n

. By

induction

, the lemma holds:

If

n

=

1

then there each element of

S

*

is distinct, and has no

intersection

with the others, so

X

=

∅

and

S

′

=

S

*

.

Suppose

n

>

1

. If there is some

x

which is in an uncountable number of elements of

S

*

then take

S

*

*

=

{

a

∖

{

x

}

∣

x

∈

a

∈

S

*

}

. Obviously this is uncountable and every

element

has

n

-

1

elements, so by the

induction hypothesis

there is some

S

′

⊆

S

*

*

of uncountable cardinality such that the intersection of any two elements is

X

. Obviously

{

a

∪

{

x

}

∣

a

∈

S

′

}

satisfies

the lemma, since the intersection of any two elements is

X

∪

{

x

}

.

On the other hand, if there is no such

x

then we can

construct

a

sequence

⟨

a

i

⟩

i

<

ω

1

such that each

a

i

∈

S

*

and for any

i

≠

j

,

a

i

∩

a

j

=

∅

by induction. Take any element for

a

0

, and given

⟨

a

i

⟩

i

<

α

, since

α

is

countable

,

A

=

⋃

i

<

α

a

i

is countable. Obviously each element of

A

is in only a countable number of elements of

S

*

, so there are an uncountable number of elements of

S

*

which are candidates for

a

α

. Then this sequence satisfies the lemma, since the intersection of any two elements is

∅

.

Title

proof of delta system lemma

Canonical

name

ProofOfDeltaSystemLemma

Date of creation

2013-03-22 12:55:03

Last modified on

2013-03-22 12:55:03

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Proof

Classification

msc 03E99

Generated on Thu Feb 8 20:21:54 2018 by

LaTeXML
properties of complement

properties of complement

Let

X

be a set and

A

,

B

are subsets of

X

.

1.

(

A

∁

)

∁

=

A

.

Proof.

a

∈

(

A

∁

)

∁

iff

a

∉

A

∁

iff

a

∈

A

.
∎

2.

∅

∁

=

X

.

Proof.

a

∈

∅

∁

iff

a

∉

∅

iff

a

∈

X

.
∎

3.

X

∁

=

∅

.

Proof.

a

∈

X

∁

iff

a

∉

X

iff

a

∈

∅

.
∎

4.

A

∪

A

∁

=

X

.

Proof.

a

∈

A

∪

A

∁

iff

a

∈

A

or

a

∈

A

∁

iff

a

∈

A

or

a

∉

A

iff

a

∈

X

.
∎

5.

A

∩

A

∁

=

∅

.

Proof.

a

∈

A

∩

A

∁

iff

a

∈

A

and

a

∈

A

∁

iff

a

∈

A

and

a

∉

A

iff

a

∈

∅

.
∎

6.

A

⊆

B

iff

B

∁

⊆

A

∁

.

Proof.

Suppose

A

⊆

B

. If

a

∈

B

∁

, then

a

∉

B

, so

a

∉

A

, or

a

∈

A

∁

. This shows that

B

∁

⊆

A

∁

. On the other hand, if

B

∁

⊆

A

∁

, then by applying what’s just been proved,

A

=

(

A

∁

)

∁

⊆

(

B

∁

)

∁

=

B

.
∎

7.

A

∩

B

=

∅

iff

A

⊆

B

∁

.

Proof.

Suppose

A

∩

B

=

∅

. If

a

∈

A

, then

a

∈

B

∁

, or

a

∉

B

, which implies that

A

∩

B

=

∅

. Suppose next that

A

⊆

B

∁

. If there is

a

∈

A

∩

B

, then

a

∈

B

and

a

∈

A

. But the second containment implies that

a

∈

B

∁

, which contradicts the first containment.
∎

8.

A

∖

B

=

A

∩

B

∁

, where the

complement

is taken in

X

.

Proof.

a

∈

A

∖

B

iff

a

∈

A

and

a

∉

B

iff

a

∈

A

and

a

∈

B

∁

iff

a

∈

A

∩

B

∁

.
∎

9.

(de Morgan’s laws)

(

A

∪

B

)

∁

=

A

∁

∩

B

∁

and

(

A

∩

B

)

∁

=

A

∁

∪

B

∁

.

Proof.

See here (

http://planetmath.org/DeMorgansLawsProof

).
∎

Title

properties of complement

Canonical name

PropertiesOfComplement

Date of creation

2013-03-22 17:55:32

Last modified on

2013-03-22 17:55:32

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 03E99

Generated on Thu Feb 8 20:22:05 2018 by

LaTeXML
saturated (set)

saturated (set)

If

p

:

X

⟶

Y

is a surjective map, we say that a subset

C

⊆

X

is

saturated

(with respect to p) if

C

contains every set

p

-

1

⁢

(

{

y

}

)

it intersects. Equivalently,

C

is saturated if it is a union of fibres.

Title

saturated (set)

Canonical

name

Saturatedset

Date of creation

2013-03-22 12:55:20

Last modified on

2013-03-22 12:55:20

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Definition

Classification

msc 03E99

Synonym

saturated

Generated on Thu Feb 8 20:22:19 2018 by

LaTeXML
superset

superset

Given two sets

A

and

B

,

A

is a

superset

of

B

if every

element

in

B

is also in

A

. We denote this

relation

as

A

⊇

B

. This is

equivalent

to saying that

B

is a subset of

A

, that is

A

⊇

B

⇔

B

⊆

A

.

Similar rules to those that hold for

⊆

also hold for

⊇

.
If

X

⊇

Y

and

Y

⊇

X

, then

X

=

Y

.
Every set is a superset of itself, and every set is a superset of the

empty set

.

We say

A

is a

proper superset

of

B

if

A

⊇

B

and

A

≠

B

. This relation is sometimes denoted by

A

⊃

B

,
but

A

⊃

B

is often used to mean the more general superset relation,
so it should be made explicit when “proper superset” is intended,
possibly by using

X

⊋

Y

or

X

⫌

Y

(or

X

⊋

Y

or

X

⫌

Y

).

One will occasionally see a

collection

C

of subsets of some set

X

made into a

partial order

“by containment”. Depending on

context

this can mean defining a partial order where

Y

≤

Z

means

Y

⊆

Z

, or it can mean defining the opposite partial order:

Y

≤

Z

means

Y

⊇

Z

. This is frequently used when applying Zorn’s lemma.

One will also occasionally see a collection

C

of subsets of some set

X

made into a

category

, usually by defining a single abstract

morphism

Y

→

Z

whenever

Y

⊆

Z

(this being a special case of the general method of treating

pre-orders

as categories). This allows a concise definition of

presheaves

and sheaves, and it is generalized when defining a site.

Title

superset

Canonical name

Superset

Date of creation

2013-05-24 14:35:12

Last modified on

2013-05-24 14:35:12

Owner

yark (2760)

Last modified by

unlord (1)

Numerical id

13

Author

yark (1)

Entry type

Definition

Classification

msc 03E99

Related topic

Subset

Related topic

SetTheory

Defines

proper superset

Defines

contains

Defines

contained

Generated on Thu Feb 8 20:22:24 2018 by

LaTeXML
Zermelo-Fraenkel axioms

Zermelo-Fraenkel axioms

Ernst Zermelo and Abraham Fraenkel proposed the following axioms as a for what is now called

Zermelo-Fraenkel set theory

, or ZF. If this set of axioms are accepted along with the

Axiom of Choice

, it is often denoted ZFC.

•

Equality of sets

: If

X

and

Y

are sets, and

x

∈

X

iff

x

∈

Y

, then

X

=

Y

.

•

Pair set

: If

X

and

Y

are sets, then there is a set

Z

containing only

X

and

Y

.

•

Union (

http://planetmath.org/Union

) over a set

: If

X

is a set, then there exists a set that contains every element of each

x

∈

X

.

•

: If

X

is a set, then there exists a set

𝒫

⁢

(

x

)

with the property that

Y

∈

𝒫

⁢

(

x

)

iff any

element

y

∈

Y

is also in

X

.

•

Replacement axiom

: Let

F

⁢

(

x

,

y

)

be some

formula

. If, for all

x

, there is exactly one

y

such that

F

⁢

(

x

,

y

)

is true, then for any set

A

there exists a set

B

with the property that

b

∈

B

iff there exists some

a

∈

A

such that

F

⁢

(

a

,

b

)

is true.

•

: Let

F

⁢

(

x

)

be some formula. If there is some

x

that makes

F

⁢

(

x

)

true, then there is a set

Y

such that

F

⁢

(

Y

)

is true, but for no

y

∈

Y

is

F

⁢

(

y

)

true.

•

Existence

of an

infinite set

: There exists a non-empty set

X

with the property that, for any

x

∈

X

, there is some

y

∈

X

such that

x

⊆

y

but

x

≠

y

.

•

: If

X

is a set and

P

is a condition on sets, there exists a set

Y

whose members are precisely the members of

X

satisfying

P

. (This axiom is also occasionally referred to as the

).

Title

Zermelo-Fraenkel axioms

Canonical name

ZermeloFraenkelAxioms

Date of creation

2013-03-22 11:47:51

Last modified on

2013-03-22 11:47:51

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

20

Author

mathcam (2727)

Entry type

Axiom

Classification

msc 03E99

Synonym

Zermelo-Fraenkel set theory

Synonym

ZFC

Synonym

ZF

Related topic

AxiomOfChoice

Related topic

RussellsParadox

Related topic

VonNeumannOrdinal

Related topic

Axiom

Related topic

ContinuumHypothesis

Related topic

GeneralizedContinuumHypothesis

Related topic

SetTheory

Related topic

VonNeumannBernausGodelSetTheory

Related topic

Set

Generated on Thu Feb 8 20:22:26 2018 by

LaTeXML