aarjaneiro/planetmath huggingface dataset:
perfect ruler

perfect ruler

A

perfect ruler

of length

n

is a ruler with a subset of the

integer

markings

{

0

,

a

2

,

…

,

n

}

⊂

{

0

,

1

,

2

,

…

,

n

}

that appear on a

regular

ruler. The defining criterion of this subset is that there exists an

m

such that any

positive integer

k

≤

m

can be expresses uniquely as a

difference

k

=

a

i

-

a

j

for some

i

,

j

. This is referred to as an

m

-perfect ruler

.

A 4-perfect ruler of length

7

is given by

{

0

,

1

,

3

,

7

}

. To verify this, we need to show that every number

1

,

2

,

…

,

4

can be expressed as a difference of two numbers in the above set:

1

=

1

-

0

2

=

3

-

1

3

=

3

-

0

4

=

7

-

3

An optimal perfect ruler is one where for a fixed value of

n

the value of

a

n

is minimized.

Title

perfect ruler

Canonical

name

PerfectRuler

Date of creation

2013-03-22 12:14:22

Last modified on

2013-03-22 12:14:22

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03E02

Classification

msc 05A17

Synonym

Golomb ruler

Generated on Thu Feb 8 20:06:47 2018 by

LaTeXML
another definition of cofinality

another definition of cofinality

Let

κ

be a

limit ordinal

(e.g. a

cardinal

). The

cofinality

of

κ

cf

⁡

(

κ

)

could also be defined as:

cf

⁡

(

κ

)

=

inf

⁡

{

|

U

|

:

U

⊆

κ

⁢

such that

⁢

sup

⁡

U

=

κ

}

(

sup

⁡

U

is calculated using the natural order of the

ordinals

).
The cofinality of a cardinal is always a regular cardinal and hence

cf

⁡

(

κ

)

=

cf

⁡

(

cf

⁡

(

κ

)

)

.

This definition is

equivalent

to the parent definition.

Title

another definition of cofinality

Canonical name

AnotherDefinitionOfCofinality

Date of creation

2013-03-22 13:52:59

Last modified on

2013-03-22 13:52:59

Owner

x_bas (2940)

Last modified by

x_bas (2940)

Numerical id

9

Author

x_bas (2940)

Entry type

Definition

Classification

msc 03E04

Generated on Thu Feb 8 20:06:49 2018 by

LaTeXML
cofinality

cofinality

Definitions

Let

(

P

,

≤

)

be a poset. A subset

A

⊆

P

is said to be

cofinal

in

P

if for every

x

∈

P

there is a

y

∈

A

such that

x

≤

y

.
A

function

f

:

X

→

P

is said to be

cofinal

if

f

⁢

(

X

)

is cofinal in

P

.
The least

cardinality

of a cofinal set of

P

is called the

cofinality

of

P

.
Equivalently, the cofinality of

P

is the least

http://planetmath.org/node/2787

ordinal

α

such that there is a cofinal function

f

:

α

→

P

.
The cofinality of

P

is written

cf

⁡

(

P

)

, or

cof

⁡

(

P

)

.

Cofinality of totally ordered sets

If

(

T

,

≤

)

is a

totally ordered set

, then it must contain a

well-ordered

cofinal subset which is order-isomorphic to

cf

⁡

(

T

)

.
Or, put another way, there is a cofinal function

f

:

cf

⁡

(

T

)

→

T

with the property that

f

⁢

(

x

)

<

f

⁢

(

y

)

whenever

x

<

y

.

For any ordinal

β

we must have

cf

⁡

(

β

)

≤

β

, because the

identity map

on

β

is cofinal.
In particular, this is true for

cardinals

, so any cardinal

κ

either

satisfies

cf

⁡

(

κ

)

=

κ

, in which case it is said to be

regular

, or it satisfies

cf

⁡

(

κ

)

<

κ

, in which case it is said to be

singular

.

The cofinality of any totally ordered set is necessarily a regular cardinal.

Cofinality of cardinals

0

and

1

are regular cardinals. All other finite cardinals have cofinality

1

and are therefore singular.

It is easy to see that

cf

⁡

(

ℵ

0

)

=

ℵ

0

, so

ℵ

0

is regular.

ℵ

1

is regular, because the union of countably many

countable sets

is

countable

.
More generally, all

infinite

successor cardinals

are regular.

The smallest infinite singular cardinal is

ℵ

ω

.
In fact, the function

f

:

ω

→

ℵ

ω

given by

f

⁢

(

n

)

=

ω

n

is cofinal, so

cf

⁡

(

ℵ

ω

)

=

ℵ

0

.
More generally, for any nonzero

limit ordinal

δ

, the function

f

:

δ

→

ℵ

δ

given by

f

⁢

(

α

)

=

ω

α

is cofinal, and this can be used to show that

cf

⁡

(

ℵ

δ

)

=

cf

⁡

(

δ

)

.

Let

κ

be an infinite cardinal.
It can be shown that

cf

⁡

(

κ

)

is
the least cardinal

μ

such that

κ

is
the sum of

μ

cardinals each of which is less than

κ

.
This fact together with König’s

theorem

tells us that

κ

<

κ

cf

⁡

(

κ

)

.
Replacing

κ

by

2

κ

in this inequality
we can further deduce that

κ

<

cf

⁡

(

2

κ

)

.
In particular,

cf

⁡

(

2

ℵ

0

)

>

ℵ

0

, from which it follows that

2

ℵ

0

≠

ℵ

ω

(this being the smallest

uncountable

aleph which is provably not the cardinality of the

continuum

).

Title

cofinality

Canonical name

Cofinality

Date of creation

2013-03-22 12:23:55

Last modified on

2013-03-22 12:23:55

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

25

Author

yark (2760)

Entry type

Definition

Classification

msc 03E04

Defines

cofinal

Defines

regular cardinal

Defines

singular cardinal

Defines

regular

Defines

singular

Generated on Thu Feb 8 20:06:51 2018 by

LaTeXML
frequently in

frequently in

Recall that a net is a

function

x

from a

directed set

D

to a set

X

. The value of

x

at

i

∈

D

is usually denoted by

x

i

. Let

A

be a subset of

X

. We say that a net

x

is

frequently in

A

if for every

i

∈

D

, there is a

j

∈

D

such that

i

≤

j

and

x

j

∈

A

.

Suppose a net

x

is frequently in

A

⊆

X

. Let

E

:=

{

j

∈

D

∣

x

j

∈

A

}

. Then

E

is a

cofinal

subset of

D

, for if

i

∈

D

, then by definition of

A

, there is

i

≤

j

∈

D

such that

x

j

∈

A

, and therefore

j

∈

E

.

The notion of “frequently in” is related to the notion of “

eventually

in” in the following sense: a net

x

is eventually in a set

A

⊆

X

iff it is not frequently in

A

∁

, its

complement

. Suppose

x

is eventually in

A

. There is

j

∈

D

such that

x

k

∈

A

for all

k

≥

j

, or equivalently,

x

k

∈

A

∁

for no

k

≥

j

. The

converse

is can be argued by tracing the previous statements backwards.

In a

topological space

X

, a point

a

∈

X

is said to be a

cluster point

of a net

x

(or, occasionally,

x

clusters at

a

) if

x

is frequently in every

neighborhood

of

a

. In this general definition, a

limit point

is always a cluster point. But a cluster point need not be a limit point. As an example, take the

sequence

0

,

2

,

0

,

4

,

0

,

6

,

0

,

8

,

…

,

0

,

2

⁢

n

,

0

,

…

has

0

as a cluster point. But clearly

0

is not a limit point, as the sequence

diverges

in

ℝ

.

Title

frequently in

Canonical name

FrequentlyIn

Date of creation

2013-03-22 17:14:23

Last modified on

2013-03-22 17:14:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E04

Synonym

clusters at

Defines

cluster point of a net

Generated on Thu Feb 8 20:07:01 2018 by

LaTeXML
maximal element

maximal element

Let

≤

be an

ordering

on a set

S

, and let

A

⊆

S

. Then, with respect to the ordering

≤

,

•

a

∈

A

is the

least

element of

A

if

a

≤

x

, for all

x

∈

A

.

•

a

∈

A

is a

minimal

element of

A

if there exists no

x

∈

A

such that

x

≤

a

and

x

≠

a

.

•

a

∈

A

is the

greatest

element of

A

if

x

≤

a

for all

x

∈

A

.

•

a

∈

A

is a

maximal

element of

A

if there exists no

x

∈

A

such that

a

≤

x

and

x

≠

a

.

Examples.

•

The

natural numbers

ℕ

ordered by

divisibility

(

∣

) have a least

element

,

1

. The natural numbers greater than 1 (

ℕ

∖

{

1

}

) have no least element, but infinitely many

minimal elements

(the primes.) In neither case is there a greatest or

maximal element

.

•

The negative integers ordered by the standard definition of

≤

have a maximal element which is also the greatest element,

-

1

. They have no minimal or least element.

•

The natural numbers

ℕ

ordered by the standard

≤

have a least element,

1

, which is also a minimal element. They have no greatest or maximal element.

•

The

rationals

greater than zero

with the standard ordering

≤

have no least element or minimal element, and no maximal or greatest element.

Title

maximal element

Canonical name

MaximalElement

Date of creation

2013-03-22 12:30:44

Last modified on

2013-03-22 12:30:44

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Definition

Classification

msc 03E04

Defines

greatest element

Defines

least element

Defines

minimal element

Generated on Thu Feb 8 20:07:03 2018 by

LaTeXML
partitions less than cofinality

partitions less than cofinality

If

λ

<

cf

⁡

(

κ

)

then

κ

→

(

κ

)

λ

1

.

This follows easily from the definition of

cofinality

. For any

coloring

f

:

κ

→

λ

then define

g

:

λ

→

κ

+

1

by

g

⁢

(

α

)

=

|

f

-

1

⁢

(

α

)

|

. Then

κ

=

∑

α

<

λ

g

⁢

(

α

)

, and by the normal rules of

cardinal arithmetic

sup

α

<

λ

⁡

g

⁢

(

α

)

=

κ

. Since

λ

<

cf

⁡

(

κ

)

, there must be some

α

<

λ

such that

g

⁢

(

α

)

=

κ

.

Title

partitions less than cofinality

Canonical name

PartitionsLessThanCofinality

Date of creation

2013-03-22 12:55:56

Last modified on

2013-03-22 12:55:56

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Result

Classification

msc 03E04

Related topic

Arrowsrelation

Related topic

ArrowsRelation

Generated on Thu Feb 8 20:07:05 2018 by

LaTeXML
visualizing maximal elements

visualizing maximal elements

Although the

mathematical definitions

of the terms
“

greatest element

” and “

maximal element

” are straightforward enough,
one may easily confuse these two terms when one first encounters them,
especially since they are not distinguished in the natural language usage
of the words. Here we give a graphical example that should
help visualize the

concept

:

Consider the complex logarithm, which is a multi-valued function on

ℂ

∖

{

0

}

. (Or equivalently, the angle function

θ

on

ℝ

2

∖

{

0

}

.)
If

U

is any simply-connected

open set

in

ℂ

∖

{

0

}

,
then it is possible to define a

single-valued

branch

f

:

U

→

ℂ

of the complex logarithm.
For example,

U

=

ℂ

∖

{

x

:

x

≤

0

}

.
This is a

maximal set

(with

set inclusion

as the

partial ordering

)
for the domain of

f

:

f

becomes

discontinuous

if we attempt to add more points to

U

.
But it is certainly not the

largest

set on which we can define

f

:
the set

U

=

ℂ

∖

{

i

⁢

y

:

y

≥

0

}

works also. Indeed, it is clearly seen that there is no such thing
as

the

largest set for the domain of

f

.
And there are obviously infinitely many maximal sets for the domain of

f

.

(Nevertheless, some people will loosely say that there is “

a

largest domain” for the single-valued branch of the logarithm.)

Title

visualizing maximal elements

Canonical

name

VisualizingMaximalElements

Date of creation

2013-03-22 15:39:36

Last modified on

2013-03-22 15:39:36

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

6

Author

stevecheng (10074)

Entry type

Example

Classification

msc 03E04

Generated on Thu Feb 8 20:07:07 2018 by

LaTeXML
cf⁡(cf⁡α)=cf⁡α

cf

⁡

(

cf

⁡

α

)

=

cf

⁡

α

Let

cf

⁡

α

=

β

and

⟨

α

ξ

:

ξ

<

β

⟩

be

cofinal

in

α

, and

cf

⁡

β

=

γ

and

⟨

ξ

(

ν

)

:

ν

<

γ

⟩

be cofinal in

β

.

The claim of the

theorem

cf

⁡

(

cf

⁡

α

)

=

cf

⁡

α

means that

γ

=

β

; we prove this fact.

Suppose

γ

≠

β

. Then

γ

<

β

by

cf

⁡

δ

≤

δ

.

Now,

⟨

α

ξ

⁢

(

ν

)

:

ν

<

γ

⟩

is seen to be confinal in

α

, which means that

cf

⁡

α

=

γ

<

β

, a

contradiction

. Therefore,

γ

=

β

.

Title

cf

⁡

(

cf

⁡

α

)

=

cf

⁡

α

Canonical name

operatornamecfoperatornamecfalphaoperatornamecfalpha

Date of creation

2013-03-22 18:11:22

Last modified on

2013-03-22 18:11:22

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

8

Author

yesitis (13730)

Entry type

Proof

Classification

msc 03E04

Generated on Thu Feb 8 20:07:09 2018 by

LaTeXML
another proof of pigeonhole principle

another proof of pigeonhole principle

By

induction

on

n

. It is harmless to let

n

=

m

+

1

, since

0

lacks

proper subsets

. Suppose that

f

:

n

→

n

is

injective

.

To begin, note that

m

∈

f

⁢

[

n

]

. Otherwise,

f

⁢

[

m

]

⊆

m

,
so that by the

induction hypothesis

,

f

⁢

[

m

]

=

m

. Then

f

⁢

[

n

]

=

f

⁢

[

m

]

, since

f

⁢

[

n

]

⊆

m

. Therefore, for some

k

<

m

,

f

⁢

(

k

)

=

f

⁢

(

m

)

.

Let

g

:

f

⁢

[

n

]

→

f

⁢

[

n

]

transpose

m

and

f

⁢

(

m

)

. Then

h

|

m

:

m

→

m

is injective, where

h

=

g

∘

f

. By the induction
hypothesis,

h

|

m

⁢

[

m

]

=

m

. Therefore:

f

⁢

[

n

]

=

g

∘

h

⁢

[

n

]

=

h

⁢

[

n

]

=

m

∪

{

m

}

=

m

+

1

=

n

⁢

.

Title

another

proof of pigeonhole principle

Canonical

name

AnotherProofOfPigeonholePrinciple

Date of creation

2013-03-22 16:02:12

Last modified on

2013-03-22 16:02:12

Owner

ratboy (4018)

Last modified by

ratboy (4018)

Numerical id

5

Author

ratboy (4018)

Entry type

Proof

Classification

msc 03E05

Generated on Thu Feb 8 20:07:11 2018 by

LaTeXML
Aronszajn tree

Aronszajn tree

A

κ

-tree (

http://planetmath.org/TreeSetTheoretic

)

T

for which

|

T

α

|

<

κ

for all

α

<

κ

and which has no

cofinal branches

is called a

κ

-

Aronszajn tree

. If

κ

=

ω

1

then it is referred to simply as an Aronszajn tree.

If there are no

κ

-Aronszajn trees for some

κ

then we say

κ

has the

tree property

.

ω

has the tree property, but no

singular cardinal

has the tree property.

Title

Aronszajn tree

Canonical

name

AronszajnTree

Date of creation

2013-03-22 12:52:34

Last modified on

2013-03-22 12:52:34

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

10

Author

Henry (455)

Entry type

Definition

Classification

msc 03E05

Classification

msc 05C05

Related topic

TreeSetTheoretic

Related topic

Antichain

Related topic

SuslinTree

Related topic

WeaklyCompactCardinalsAndTheTreeProperty

Defines

Aronszajn tree

Defines

κ

-Aronszajn tree

Defines

tree property

Generated on Thu Feb 8 20:07:13 2018 by

LaTeXML
example of Aronszajn tree

example of Aronszajn tree

Construction 1:

If

κ

is a

singular cardinal

then there is a construction of a

κ

-

Aronszajn tree

(

http://

planetmath

.org/KappaAronszjanTree

). Let

⟨

k

β

⟩

β

<

ι

with

ι

<

κ

be a

sequence

cofinal in

κ

. Then consider the tree where

T

=

{

(

α

,

k

β

)

∣

α

<

k

β

∧

β

<

ι

}

with

(

α

1

,

k

β

1

)

<

T

(

α

2

,

k

β

2

)

iff

α

1

<

α

2

and

k

β

1

=

k

β

2

.

Note that this is

similar

to (indeed, a

subtree

of) the construction given for a tree with no

cofinal branches

. It consists of

ι

disjoint

branches, with the

β

-th branch of height

k

β

. Since

ι

<

κ

, every level has fewer than

κ

elements

, and since the sequence is cofinal in

κ

,

T

must have height and

cardinality

κ

.

Construction 2:

We can construct an Aronszajn tree out of the compact subsets of

ℚ

+

.

<

T

will be defined by

x

<

T

y

iff

y

is an end-extension of

x

. That is,

x

⊆

y

and if

r

∈

y

∖

x

and

s

∈

x

then

s

<

r

.

Let

T

0

=

{

[

0

]

}

. Given a level

T

α

, let

T

α

+

1

=

{

x

∪

{

q

}

⁢

∣

x

∈

T

α

∧

q

>

⁢

max

⁡

x

}

. That is, for every element

x

in

T

α

and every

rational number

q

larger than any element of

x

,

x

∪

{

q

}

is an element of

T

α

+

1

. If

α

<

ω

1

is a

limit ordinal

then each element of

T

α

is the union of some branch in

T

⁢

(

α

)

.

We can show by

induction

that

|

T

α

|

<

ω

1

for each

α

<

ω

1

. For the case,

T

0

has only one element. If

|

T

α

|

<

ω

1

then

|

T

α

+

1

|

=

|

T

α

|

⋅

|

ℚ

|

=

|

T

α

|

⋅

ω

=

ω

<

ω

1

. If

α

<

ω

1

is a limit ordinal then

T

⁢

(

α

)

is a

countable

union of countable sets

, and therefore itself countable. Therefore there are a countable number of branches, so

T

α

is also countable. So

T

has countable levels.

Suppose

T

has an

uncountable

branch,

B

=

⟨

b

0

,

b

1

,

…

⟩

. Then for any

i

<

j

<

ω

1

,

b

i

⊂

b

j

. Then for each

i

, there is some

x

i

∈

b

i

+

1

∖

b

i

such that

x

i

is greater than any element of

b

i

. Then

⟨

x

0

,

x

1

,

…

⟩

is an uncountable

increasing sequence

of rational numbers. Since the rational numbers are countable, there is no such sequence, so

T

has no uncountable branch, and is therefore Aronszajn.

Title

example of Aronszajn tree

Canonical name

ExampleOfAronszajnTree

Date of creation

2013-03-22 12:52:39

Last modified on

2013-03-22 12:52:39

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Example

Classification

msc 03E05

Classification

msc 05C05

Generated on Thu Feb 8 20:07:16 2018 by

LaTeXML
ordered tree

ordered tree

A tree is, by definition, a

partially ordered set

. An

ordered tree

is a tree equipped with a

linear extension

. In other words, its a

linear order

on the nodes of the tree that obeys the parent-child relationship of the tree. By Zorn’s lemma, every tree can be ordered. When a tree

T

is finite, the linear extension of the tree can be constructed inductively:

1.

First, label the nodes of

T

: each label is an

n

-tuple of positive integers, for some

n

,

(a)

the root of

T

is labeled

(

1

)

,

(b)

if a node

V

of

T

has been labeled

(

m

1

,

…

,

m

k

)

, and if the children of

T

are

V

1

,

…

,

V

ℓ

, then label each

V

i

by

(

m

1

,

…

,

m

k

,

i

)

.

From this

labeling

process, we see that the (direct) children of the root have labels

(

1

,

1

)

,

(

1

,

2

)

,

…

. In general, if a node is of level

k

, its label is a

k

-tuple starting with

1

. It is easy to see that each label corresponds to a unique node of

T

: for if two distinct nodes are of different level, their labels are clearly different. However, if they are of the same level, then the last

components

of their labels are distinct.

2.

Next, order the labels of the nodes lexicographically. In other words, given an

m

-tuple

p

=

(

p

1

,

…

,

p

m

)

and an

n

-tuple

q

=

(

q

1

,

…

,

q

n

)

, compare them componentwise, starting from the first. There are two cases:

(a)

either all

k

components match, where

k

=

min

⁡

(

m

,

n

)

, or

(b)

p

j

≠

q

j

, but

p

i

=

q

i

for all

i

<

j

, where

j

≤

min

⁡

(

m

,

n

)

.

Then

p

<

q

iff

m

<

n

in the first case, or

p

j

<

q

j

in the second. This

ordering

on the labels induces an ordering on the nodes: for nodes

V

and

V

′

with labels

p

and

p

′

,

V

≤

V

′

iff

p

≤

p

′

.

This ordering on the nodes is linear, and extends the partial ordering on

T

.

For example, below are two diagrams of the same tree. The one on the right is labeled by the method above:

By ordering the labels, we get the corresponding ordering of the nodes:

a

<

b

<

e

<

f

<

c

<

g

<

h

<

i

<

j

<

k

<

d

.

Title

ordered tree

Canonical name

OrderedTree

Date of creation

2013-03-22 19:00:11

Last modified on

2013-03-22 19:00:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E05

Generated on Thu Feb 8 20:07:27 2018 by

LaTeXML
pigeonhole principle

pigeonhole principle

For any

natural number

n

, there does not exist a

bijection

between

n

and a

proper subset

of

n

.

The name of the

theorem

is based upon the observation that pigeons will not occupy a pigeonhole that already contains a pigeon, so there is no way to fit

n

pigeons in fewer than

n

pigeonholes.

Title

pigeonhole principle

Canonical name

PigeonholePrinciple

Date of creation

2013-03-22 11:53:32

Last modified on

2013-03-22 11:53:32

Owner

djao (24)

Last modified by

djao (24)

Numerical id

11

Author

djao (24)

Entry type

Theorem

Classification

msc 03E05

Classification

msc 03B22

Classification

msc 03-01

Classification

msc 03-00

Synonym

box principle

Synonym

Dirichlet principle

Generated on Thu Feb 8 20:07:28 2018 by

LaTeXML
proof of pigeonhole principle

proof of pigeonhole principle

Proof.

It will first be proven that, if a

bijection

exists between two

finite sets

, then the two sets have the same number of

elements

. Let

S

and

T

be finite sets and

f

:

S

→

T

be a bijection. The claim will be proven by

induction

on

|

S

|

.

If

|

S

|

=

0

, then

S

=

∅

, and

f

:

∅

→

T

can only be

surjective

if

T

=

∅

.

Assume the statement holds for any set

S

with

|

S

|

=

n

. Let

|

S

|

=

n

+

1

. Let

x

1

,

…

,

x

n

+

1

∈

S

with

S

=

{

x

1

,

…

,

x

n

+

1

}

. Let

R

=

S

∖

{

x

n

+

1

}

. Then

|

R

|

=

n

.

Define

g

:

R

→

T

∖

{

f

⁢

(

x

n

+

1

)

}

by

g

⁢

(

x

)

=

f

⁢

(

x

)

. Since

R

⊂

S

,

f

⁢

(

x

)

∈

T

for all

x

∈

R

. Thus, to show that

g

is

well-defined

, it only needs to be verified that

f

⁢

(

x

)

≠

f

⁢

(

x

n

+

1

)

for all

x

∈

R

. This follows immediately from the facts that

x

n

+

1

∉

R

and

f

is

injective

. Therefore,

g

is well-defined.

Now it need to be proven that

g

is a bijection. The fact that

g

is injective follows immediately from the fact that

f

is injective. To verify that

g

is surjective, let

y

∈

T

∖

{

f

⁢

(

x

n

+

1

)

}

. Since

f

is surjective, there exists

x

∈

S

with

f

⁢

(

x

)

=

y

. Since

f

⁢

(

x

)

=

y

≠

f

⁢

(

x

n

+

1

)

and

f

is injective,

x

≠

x

n

+

1

. Thus,

x

∈

R

. Hence,

g

⁢

(

x

)

=

f

⁢

(

x

)

=

y

. It follows that

g

is a bijection.

By the

induction hypothesis

,

|

R

|

=

|

T

∖

{

f

⁢

(

x

n

+

1

)

}

|

. Thus,

n

=

|

R

|

=

|

T

∖

{

f

⁢

(

x

n

+

1

)

}

|

=

|

T

|

-

1

. Therefore,

|

T

|

=

n

+

1

=

|

S

|

.
∎

Title

proof of pigeonhole principle

Canonical

name

ProofOfPigeonholePrinciple

Date of creation

2013-03-22 13:31:09

Last modified on

2013-03-22 13:31:09

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 03E05

Generated on Thu Feb 8 20:07:32 2018 by

LaTeXML