aarjaneiro/planetmath huggingface dataset:
example of uncountable family of subsets of a countable set with finite intersections

example of uncountable family of subsets of a countable set with finite intersections

We wish to give an answer to the following:

Problem.

Assume, that

X

is a

countable set

. Is there a family

{

X

i

}

i

∈

I

of subsets of

X

such that

I

is an

uncountable set

, but for any

i

≠

j

∈

I

the

intersection

X

i

∩

X

j

is finite?

Example.

Let

x

∈

[

1

,

2

)

be a

real number

. Express

x

using digits

x

=

1

.

x

1

⁢

x

2

⁢

x

3

⁢

x

4

⁢

⋯

=

1

+

∑

i

=

1

∞

x

i

⋅

10

-

i

where each

x

i

∈

{

0

,

1

,

2

,

3

,

4

,

5

,

6

,

7

,

8

,

9

}

. With

x

we associate the following

natural numbers

β

n

⁢

(

x

)

=

1

⁢

x

1

⁢

x

2

⁢

x

3

⁢

⋯

⁢

x

n

-

1

⁢

x

n

=

10

n

+

1

+

∑

i

=

1

n

x

i

⋅

10

n

-

i

+

1

.

Now define

A

:

[

1

,

2

)

→

P

⁢

(

ℕ

)

(here

P

⁢

(

X

)

stands for ,,the

power set

of

X

”) by

A

⁢

(

x

)

=

{

β

1

⁢

(

x

)

,

β

2

⁢

(

x

)

,

β

3

⁢

(

x

)

,

…

}

.

A

is

injective

. Indeed, note that for any

x

,

y

∈

[

1

,

2

)

if

β

i

⁢

(

x

)

=

β

j

⁢

(

y

)

, then

i

=

j

(this is because equal

β

numbers have equal ,,length” and this is because each

β

has

1

at the begining, zeros are not the problem). Therefore, if

A

⁢

(

x

)

=

A

⁢

(

y

)

for some

x

,

y

, then it follows, that

β

i

⁢

(

x

)

=

β

i

⁢

(

y

)

for each

i

, but this implies that corresponding digits of

x

and

y

are equal. Thus

x

=

y

.

This shows, that

{

A

⁢

(

x

)

}

x

∈

[

1

,

2

)

is an uncountable family of subsets of

ℕ

. Now in order to prove that

A

⁢

(

x

)

∩

A

⁢

(

y

)

is finite whenever

x

≠

y

it is enough to show that we can uniquely reconstruct

x

from any

infinite

sequence

of numbers from

A

⁢

(

x

)

. This can be proved by using

similar

techniques as before and we leave it as a simple exercise.

Title

example of uncountable family of subsets of a countable set with finite intersections

Canonical name

ExampleOfUncountableFamilyOfSubsetsOfACountableSetWithFiniteIntersections

Date of creation

2013-03-22 19:16:29

Last modified on

2013-03-22 19:16:29

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Example

Classification

msc 03E10

Generated on Thu Feb 8 20:08:59 2018 by

LaTeXML
finite

finite

A set

S

is

finite

if there exists a

natural number

n

and a

bijection

from

S

to

n

. Note that we are using the set theoretic definition of natural number, under which the natural number

n

equals the set

{

0

,

1

,

2

,

…

,

n

-

1

}

. If there exists such an

n

, then it is unique, and we call

n

the

cardinality

of

S

.

Equivalently, a set

S

is finite if and only if there is no bijection between

S

and any

proper subset

of

S

.

Title

finite

Canonical name

Finite

Date of creation

2013-03-22 11:53:25

Last modified on

2013-03-22 11:53:25

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 03E10

Classification

msc 92C05

Classification

msc 92B05

Classification

msc 18-00

Classification

msc 92C40

Classification

msc 18-02

Related topic

Infinite

Defines

finite set

Generated on Thu Feb 8 20:09:11 2018 by

LaTeXML
fixed points of normal functions

fixed points of normal functions

If

f

:

M

→

𝐎𝐧

is a

function

from any set of

ordinals

to the class of ordinals then

Fix

⁡

(

f

)

=

{

x

∈

M

∣

f

⁢

(

x

)

=

x

}

is the set of

fixed points

of

f

.

f

′

, the

derivative

of

f

, is the

enumerating function

of

Fix

⁡

(

f

)

.

If

f

is

κ

-normal (

http://

planetmath

.org/KappaNormal

) then

Fix

⁡

(

f

)

is

κ

-closed and

κ

-normal, and therefore

f

′

is also

κ

-normal.

For example, the function which takes an ordinal

α

to the ordinal

1

+

α

has a fixed point at every ordinal

≥

ω

, so

f

′

⁢

(

α

)

=

ω

+

α

.

Title

fixed points of normal functions

Canonical

name

FixedPointsOfNormalFunctions

Date of creation

2013-03-22 13:28:59

Last modified on

2013-03-22 13:28:59

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 03E10

Related topic

ProofOfPowerRule

Related topic

LeibnizNotation

Related topic

ProofOfProductRule

Related topic

ProofOfSumRule

Related topic

SumRule

Related topic

DirectionalDerivative

Related topic

NewtonsMethod

Defines

derivative

Generated on Thu Feb 8 20:09:13 2018 by

LaTeXML
Fodor’s lemma

Fodor’s lemma

If

κ

is a

regular

,

uncountable

cardinal

,

S

is a

stationary

subset of

κ

, and

f

:

κ

→

κ

is regressive on

S

(that is,

f

⁢

(

α

)

<

α

for any

α

∈

S

) then there is some

γ

and some stationary

S

0

⊆

S

such that

f

⁢

(

α

)

=

γ

for any

α

∈

S

0

.

Title

Fodor’s lemma

Canonical name

FodorsLemma

Date of creation

2013-03-22 12:53:14

Last modified on

2013-03-22 12:53:14

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Theorem

Classification

msc 03E10

Synonym

pushing down lemma

Related topic

Stationary

Defines

Fodor’s lemma

Generated on Thu Feb 8 20:09:15 2018 by

LaTeXML
Galileo’s paradox

Galileo’s paradox

Galileo Galilei (1564—1642) has realised the ostensible

contradiction

in the situation, that although the set

1

,

2

,

3

,

4

,

5

,

…

of the

positive integers

all the members of the set

1

,

4

,

9

,

16

,

…

of the

perfect squares

and in many others, however both sets are equally great in the sense that any member of the former set has as its square a unique counterpart in the latter set and also any member of the latter set has as its

square root

a unique counterpart in the former set.  Galileo explained this by the infinitude of the sets.

In modern mathematical , we say that an

infinite set

and its

proper subset

set may have the same

cardinality

.

Title

Galileo’s

paradox

Canonical name

GalileosParadox

Date of creation

2013-03-22 19:15:45

Last modified on

2013-03-22 19:15:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Definition

Classification

msc 03E10

Related topic

Paradox

Generated on Thu Feb 8 20:09:17 2018 by

LaTeXML
Golomb’s sequence

Golomb’s sequence

Golomb’s

sequence

is a self-referential ascending order sequence of

integers

a

in which the number of times each number

n

occurrs is given by

a

n

. The sequence begins 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, … (see A001462 in Sloane’s OEIS). For example,

a

3

=

2

and indeed 3 occurs twice in the sequence.

a

4

=

3

, so 4 occurs thrice.

The

recurrence relation

for the sequence is

a

1

=

1

and

a

n

=

a

n

-

a

a

a

n

-

1

+

1

. Sometimes recurrence relations for this sequence are given which also explicitly set

a

2

=

2

. The

n

th term of the sequence can be obtained by the calculating

ϕ

2

-

ϕ

⁢

n

ϕ

-

1

(with

ϕ

=

1

+

5

2

being the

golden ratio

) and

rounding

off to the nearest integer. For example, with

precision

to five

decimal places

: 1.20178, 1.84448, 2.36975, 2.83087, 3.24948, 3.63706, 4.0006, 4.34477, 4.67284, 4.98724, 5.28984, 5.58209, 5.86518, 6.14005, 6.40753, 6.66827, 6.92286, 7.17178, 7.41548, etc.

Title

Golomb’s sequence

Canonical

name

GolombsSequence

Date of creation

2013-03-22 17:47:07

Last modified on

2013-03-22 17:47:07

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 03E10

Synonym

Golomb sequence

Synonym

Silverman’s sequence

Synonym

Silverman sequence

Generated on Thu Feb 8 20:09:19 2018 by

LaTeXML
height of an algebraic number

height of an algebraic number

Suppose we have an

algebraic number

such that the

polynomial

of smallest degree it is a root of (with the co-efficients relatively prime) is given by:

∑

i

=

0

n

a

i

⁢

x

i

.

Then the height

h

of the algebraic number is given by:

h

=

n

+

∑

i

=

0

n

|

a

i

|

.

This is a quantity which is used in the proof of the existence of

transcendental numbers

.

References

1

Shaw, R. Mathematics Society Notes, 1st edition. King’s School Chester, 2003.

2

Stewart, I.

Galois Theory

, 3rd edition. Chapman and Hall, 2003.

3

Baker, A. Transcendental Number Theory, 1st edition. Cambridge University Press, 1975.

Title

height of an algebraic number

Canonical name

HeightOfAnAlgebraicNumber

Date of creation

2013-03-22 13:24:34

Last modified on

2013-03-22 13:24:34

Owner

kidburla2003 (1480)

Last modified by

kidburla2003 (1480)

Numerical id

17

Author

kidburla2003 (1480)

Entry type

Definition

Classification

msc 03E10

Synonym

height

Related topic

AlgebraicNumbersAreCountable

Generated on Thu Feb 8 20:09:31 2018 by

LaTeXML
Hilbert’s hotel

Hilbert’s hotel

The hotel manager

David Hilbert

had a very large hotel, in fact, it had infinitely many rooms numbered

1

,

2

,

3

,

…

. The hotel was very popular and every room was occupied. One day a new guest arrived.

-Is there any free room?

-No, Mr. Hilbert said.

-Oh, what a pity, the guest said and started to walk away.

-But wait, you can still get a room.

The new guest was very confused by this and asked how that could be possible.

-I’ll just ask the guest in room number 1 to move to room number 2, the guest in number 2 move to room 3, the guest in room 3 move to number 4, and so on, and then you can have room number 1.

The guest was very happy with this and called his friends to tell them about this fantastic hotel. Then one day they all arrived at the same time.

-Hello, we are countably (

http://

planetmath

.org/

Countable

) many people, and we want a room each.
Mr. Hilbert felt reluctant to ask each guest to move to a new room an

infinite

number of times. That would be very unpleasant, and they will never finish either. Luckily he got a brilliant idea.

-I’ll let guest in number 1 move into number 2, guest in number 2 move into number 4, guest in number 3 move into number 6, number 4 to number 8 and so on.
Then you can move into any room with an

odd number

; they will all be free, and there will be rooms for all of you.

Now all the new guests got a room on their own.

Historical note:

David Hilbert is one of the great mathematicians in the history of mathematics. One of the things he studied was the

foundations of mathematics

and that includes the nature of

infinity

.

Title

Hilbert’s hotel

Canonical name

HilbertsHotel

Date of creation

2013-03-22 14:18:55

Last modified on

2013-03-22 14:18:55

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

11

Author

Daume (40)

Entry type

Topic

Classification

msc 03E10

Generated on Thu Feb 8 20:09:33 2018 by

LaTeXML
if A is infinite and B is a finite subset of A⁢ERROR(\tmspace)+.1667⁢e⁢m⁢ERROR(\tmspace)-.1667⁢e⁢m, then A∖B is infinite

if

A

is infinite and

B

is a finite subset of

A

⁢

\tmspace

+

.1667

⁢

e

⁢

m

⁢

\tmspace

-

.1667

⁢

e

⁢

m

,

then

A

∖

B

is infinite

Theorem

.

If

A

is an

infinite set

and

B

is a
finite subset of

A

, then

A

∖

B

is

infinite

.

Proof.

The proof is by

contradiction

. If

A

∖

B

would be finite, there would exist a

k

∈

ℕ

and a

bijection

f

:

{

1

,

…

,

k

}

→

A

∖

B

. Since

B

is finite, there
also exists a bijection

g

:

{

1

,

…

,

l

}

→

B

. We can then define
a

mapping

h

:

{

1

,

…

,

k

+

l

}

→

A

by

h

⁢

(

i

)

=

{

f

⁢

(

i

)

when

⁢

i

∈

{

1

,

…

,

k

}

,

g

⁢

(

i

-

k

)

when

⁢

i

∈

{

k

+

1

,

…

,

k

+

l

}

.

Since

f

and

g

are bijections,

h

is a bijection between
a finite subset of

ℕ

and

A

. This is a contradiction
since

A

is infinite.

□

Title

if

A

is infinite and

B

is a finite subset of

A

⁢

\tmspace

+

.1667

⁢

e

⁢

m

⁢

\tmspace

-

.1667

⁢

e

⁢

m

,

then

A

∖

B

is infinite

Canonical name

IfAIsInfiniteAndBIsAFiniteSubsetOfAThenAsetminusBIsInfinite

Date of creation

2013-03-22 13:34:42

Last modified on

2013-03-22 13:34:42

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 03E10

Generated on Thu Feb 8 20:09:45 2018 by

LaTeXML
intercession

intercession

The

intercession

is an alternative measure for

infinite sets

of finite numbers.

Definition

: Two infinite sets,

A

and

B

,

intercede

(each other) if they can be put in an

intercession

, i.e., if they can be ordered such that

A

is dense in

B

and

B

is dense in

A

. In other words, between two elements of

A

there is at least one element of

B

and,

vice versa

, between two elements of

B

there is at least one element of

A

.

The intercession of sets with nonempty

intersection

, e.g., the intercession of a set with itself, requires distinction of identical

elements

. As an example an intercession of the set of positive integers and the set of even positive integers, 1, 2, 3, … and 2’, 4’, 6’, …, is given by 1, 2’, 2, 4’, 3, 6’, ….

The intercession includes Cantor’s definition of

equivalent

(or

equipotent

) sets by

one-to-one correspondence

(or bijection): Two equivalent sets always intercede each other, i.e., they can always be put in an intercession. The intercession is an

equivalence relation

. All infinite sets of finite numbers (like the integers, the

rationals

or the reals) belong, under this

relation

, to the same

equivalence class

.

Literature

W. Mückenheim: Die Mathematik des Unendlichen, Shaker, Aachen 2006.

Title

intercession

Canonical

name

Intercession

Date of creation

2013-03-22 17:02:23

Last modified on

2013-03-22 17:02:23

Owner

WM (16977)

Last modified by

WM (16977)

Numerical id

10

Author

WM (16977)

Entry type

Definition

Classification

msc 03E10

Classification

msc 03E70

Synonym

intercede

Generated on Thu Feb 8 20:09:47 2018 by

LaTeXML
König’s theorem

König’s theorem

König’s

Theorem

is a theorem of

cardinal arithmetic

.

Theorem 1

.

Let

κ

i

and

λ

i

be

cardinals

, for all

i

in some

index set

I

.
If

κ

i

<

λ

i

for all

i

∈

I

, then

∑

i

∈

I

κ

i

<

∏

i

∈

I

λ

i

.

The theorem can also be stated for arbitrary sets, as follows.

Theorem 2

.

Let

A

i

and

B

i

be sets, for all

i

in some index set

I

.
If

|

A

i

|

<

|

B

i

|

for all

i

∈

I

, then

|

⋃

i

∈

I

A

i

|

<

|

∏

i

∈

I

B

i

|

.

Proof.

Let

φ

:

⋃

i

∈

I

A

i

→

∏

i

∈

I

B

i

be a

function

.
For each

i

∈

I

we have

|

φ

⁢

(

A

i

)

|

≤

|

A

i

|

<

|

B

i

|

,
so there is some

x

i

∈

B

i

that is not equal to

(

φ

⁢

(

a

)

)

⁢

(

i

)

for any

a

∈

A

i

.
Define

f

:

I

→

⋃

i

∈

I

B

i

by

f

⁢

(

i

)

=

x

i

for all

i

∈

I

.
For any

i

∈

I

and any

a

∈

A

i

,
we have

f

⁢

(

i

)

≠

(

φ

⁢

(

a

)

)

⁢

(

i

)

, so

f

≠

φ

⁢

(

a

)

.
Therefore

f

is not in the image of

φ

.
This shows that there is
no

surjection

from

⋃

i

∈

I

A

i

onto

∏

i

∈

I

B

i

.
As

∏

i

∈

I

B

i

is nonempty,
this also means that
there is no

injection

from

∏

i

∈

I

B

i

into

⋃

i

∈

I

A

i

.
This

completes

the proof of Theorem 2.
Theorem 1 follows as an immediate corollary.
∎

Note that the above proof is a diagonal argument,

similar

to the proof of Cantor’s Theorem.
In fact, Cantor’s Theorem can be considered as a special case of König’s Theorem, taking

κ

i

=

1

and

λ

i

=

2

for all

i

.

Also note that Theorem 2 is

equivalent

(in ZF) to the

Axiom of Choice

, as it implies that

products

(

http://planetmath.org/GeneralizedCartesianProduct

) of

nonempty sets

are nonempty. (Theorem 1, on the other hand, is not meaningful without the Axiom of Choice.)

Title

König’s theorem

Canonical name

KonigsTheorem

Date of creation

2013-03-22 14:10:21

Last modified on

2013-03-22 14:10:21

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Theorem

Classification

msc 03E10

Synonym

Koenig’s theorem

Synonym

Konig’s theorem

Synonym

König-Zermelo theorem

Synonym

Koenig-Zermelo theorem

Synonym

Konig-Zermelo theorem

Related topic

CantorsTheorem

Generated on Thu Feb 8 20:09:50 2018 by

LaTeXML
limit cardinal

limit cardinal

A

limit cardinal

is a

cardinal

κ

such that

λ

+

<

κ

for every cardinal

λ

<

κ

. Here

λ

+

denotes the

cardinal successor

of

λ

. If

2

λ

<

κ

for every cardinal

λ

<

κ

, then

κ

is called a

strong limit cardinal

.

Every strong limit cardinal is a limit cardinal, because

λ

+

≤

2

λ

holds for every cardinal

λ

.
Under GCH, every limit cardinal is a strong limit cardinal because in this case

λ

+

=

2

λ

for every

infinite

cardinal

λ

.

The three smallest limit cardinals are

0

,

ℵ

0

and

ℵ

ω

.
Note that some authors do not count

0

, or sometimes even

ℵ

0

, as a limit cardinal.
An infinite cardinal

ℵ

α

is a limit cardinal
if and only if

α

is either

0

or a

limit ordinal

.

Title

limit cardinal

Canonical name

LimitCardinal

Date of creation

2013-03-22 14:04:40

Last modified on

2013-03-22 14:04:40

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Definition

Classification

msc 03E10

Related topic

SuccessorCardinal

Defines

strong limit cardinal

Generated on Thu Feb 8 20:09:52 2018 by

LaTeXML
natural number

natural number

Given the

Zermelo-Fraenkel axioms

of

set theory

, one can prove that there exists an

inductive set

X

such that

∅

∈

X

. The

natural numbers

ℕ

are then defined to be the

intersection

of all subsets of

X

which are inductive sets and contain the

empty set

as an

element

.

The first few natural numbers are:

•

0

:=

∅

•

1

:=

0

′

=

{

0

}

=

{

∅

}

•

2

:=

1

′

=

{

0

,

1

}

=

{

∅

,

{

∅

}

}

•

3

:=

2

′

=

{

0

,

1

,

2

}

=

{

∅

,

{

∅

}

,

{

∅

,

{

∅

}

}

}

Note that the set

0

has

zero elements

, the set

1

has one element, the set

2

has two elements, etc. Informally, the set

n

is the set consisting of the

n

elements

0

,

1

,

…

,

n

-

1

, and

n

is both a subset of

ℕ

and an element of

ℕ

.

In some

contexts

(most notably, in

number theory

), it is more convenient to exclude

0

from the set of natural numbers, so that

ℕ

=

{

1

,

2

,

3

,

…

}

. When it is not explicitly specified, one must determine from context whether

0

is being considered a natural number or not.

Addition

of natural numbers is defined inductively as follows:

•

a

+

0

:=

a

for all

a

∈

ℕ

•

a

+

b

′

:=

(

a

+

b

)

′

for all

a

,

b

∈

ℕ

Multiplication

of natural numbers is defined inductively as follows:

•

a

⋅

0

:=

0

for all

a

∈

ℕ

•

a

⋅

b

′

:=

(

a

⋅

b

)

+

a

for all

a

,

b

∈

ℕ

The natural numbers form a monoid under either addition or multiplication. There is an

ordering relation

on the natural numbers, defined by:

a

≤

b

if

a

⊆

b

.

Title

natural number

Canonical

name

NaturalNumber

Date of creation

2013-03-22 11:50:05

Last modified on

2013-03-22 11:50:05

Owner

djao (24)

Last modified by

djao (24)

Numerical id

16

Author

djao (24)

Entry type

Definition

Classification

msc 03E10

Classification

msc 74D99

Synonym

ℕ

Related topic

InductiveSet

Related topic

Successor

Related topic

PeanoArithmetic

Related topic

VonNeumannInteger

Generated on Thu Feb 8 20:09:59 2018 by

LaTeXML
natural numbers are well-ordered

natural numbers are well-ordered

In many proofs, one needs the following property of

positive

and

nonnegative integers

:

Theorem

.

Any non-empty set of natural numbers contains a least number.

Proof.

Let

A

be an arbitrary non-empty subset of

ℕ

.  Denote

C

=

{

x

∈

ℕ

⋮

x

≤

a

∀

a

∈

A

}

.

Then of course,

0

∈

C

.  There exists surely an

element

c

of

C

such that

c

+

1

∉

C

,  since otherwise the

induction

property would imply that

C

=

ℕ

.  Because

c

+

1

∉

C

,  there is a number

a

0

of the set

A

such that

a

0

<

c

+

1

.  On the other , we must have

c

≤

a

0

.  Consequently,

c

=

a

0

and therefore

a

0

=

c

≤

a

⁢

∀

a

∈

A

.

Hence,

A

has the least number

a

0

.  Q.E.D.

Title

natural numbers are well-ordered

Canonical name

NaturalNumbersAreWellordered

Date of creation

2013-03-22 19:02:36

Last modified on

2013-03-22 19:02:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 03E10

Related topic

AVariantDerivationOfWellOrderedSet

Related topic

WellOrderedSet

Related topic

WellOrderingPrincipleForNaturalNumbersProvenFromThePrincipleOfFiniteInduction

Generated on Thu Feb 8 20:10:01 2018 by

LaTeXML
normal (ordinal) function

normal (ordinal) function

Definition

.

A

function

F

:

𝐎𝐧

→

𝐎𝐧

is

continuous

if and only if for each

u

⊂

𝐎𝐧

such that

u

≠

∅

it holds that

F

⁢

(

sup

⁡

(

u

)

)

=

sup

⁡

{

F

⁢

(

α

)

|

α

∈

u

}

.

Definition

.

A function

F

:

𝐎𝐧

→

𝐎𝐧

is

order preserving

if and only if for each

α

,

β

∈

𝐎𝐧

such that

α

<

β

it follows that

F

⁢

(

α

)

<

F

⁢

(

β

)

.

Definition

.

A function

F

:

𝐎𝐧

→

𝐎𝐧

is a

normal

function if and only if

F

is continuous and order preserving.

Title

normal (ordinal) function

Canonical name

NormalordinalFunction

Date of creation

2013-03-22 15:33:10

Last modified on

2013-03-22 15:33:10

Owner

florisje (7763)

Last modified by

florisje (7763)

Numerical id

7

Author

florisje (7763)

Entry type

Definition

Classification

msc 03E10

Defines

continuous (for ordinal functions)

Defines

order preserving (for ordinal functions)

Defines

normality

Defines

normal function

Generated on Thu Feb 8 20:10:03 2018 by

LaTeXML