aarjaneiro/planetmath huggingface dataset:
a surjection between finite sets of the same cardinality is bijective

a surjection between finite sets of the same cardinality is bijective

Theorem

.

Let

A

and

B

be

finite sets

of the same

cardinality

. If

f

:

A

→

B

is a

surjection

then

f

is a

bijection

.

Proof.

Let

A

and

B

be finite sets with

|

A

|

=

|

B

|

=

n

. Let

C

=

{

f

-

1

⁢

(

{

b

}

)

∣

b

∈

B

}

. Then

⋃

C

⊆

A

, so

|

⋃

C

|

≤

n

. Since

f

is a surjection,

|

f

-

1

⁢

(

{

b

}

)

|

≥

1

for each

b

∈

B

. The sets in

C

are

pairwise disjoint

because

f

is a

function

; therefore,

n

≤

|

⋃

C

|

and

|

⋃

C

|

=

∑

b

∈

B

|

f

-

1

⁢

(

{

b

}

)

|

.

In the last equation,

n

has been expressed as
the sum of

n

positive integers

; thus

|

f

-

1

⁢

(

{

b

}

)

|

=

1

for each

b

∈

B

, so

f

is

injective

.
∎

Title

a surjection between finite sets of the same cardinality is

bijective

Canonical name

ASurjectionBetweenFiniteSetsOfTheSameCardinalityIsBijective

Date of creation

2013-03-22 15:23:28

Last modified on

2013-03-22 15:23:28

Owner

ratboy (4018)

Last modified by

ratboy (4018)

Numerical id

5

Author

ratboy (4018)

Entry type

Result

Classification

msc 03-00

Related topic

OneToOneFunctionFromOntoFunction

Generated on Thu Feb 8 19:18:21 2018 by

LaTeXML
an injection between two finite sets of the same cardinality is bijective

an injection between two finite sets of the same cardinality is bijective

Lemma

.

Let

A

,

B

be two

finite sets

of the same

cardinality

. If

f

:

A

→

B

is an

injective function

then

f

is

bijective

.

Proof.

In order to prove the lemma, it suffices to show that if

f

is an injection then the cardinality of

f

⁢

(

A

)

and

A

are equal. We prove this by

induction

on

n

=

card

⁢

(

A

)

. The case

n

=

1

is

trivial

. Assume that the lemma is true for sets of cardinality

n

and let

A

be a set of cardinality

n

+

1

. Let

a

∈

A

so that

A

1

=

A

-

{

a

}

has cardinality

n

. Thus,

f

⁢

(

A

1

)

has cardinality

n

by the

induction hypothesis

. Moreover,

f

⁢

(

a

)

∉

f

⁢

(

A

1

)

because

a

∉

A

1

and

f

is injective. Therefore:

f

⁢

(

A

)

=

f

⁢

(

{

a

}

∪

A

1

)

=

{

f

⁢

(

a

)

}

∪

f

⁢

(

A

1

)

and the set

{

f

⁢

(

a

)

}

∪

f

⁢

(

A

1

)

has cardinality

1

+

n

, as desired.
∎

Title

an injection between two finite sets of the same cardinality is bijective

Canonical

name

AnInjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective

Date of creation

2013-03-22 15:10:20

Last modified on

2013-03-22 15:10:20

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 03-00

Related topic

SchroederBernsteinTheorem

Related topic

ProofOfSchroederBernsteinTheorem

Related topic

OneToOneFunctionFromOntoFunction

Generated on Thu Feb 8 19:18:23 2018 by

LaTeXML
biconditional

biconditional

1

Biconditional

A

biconditional

is a

truth function

that is true only in the case that both

parameters

are true or both are false.

Symbolically, the biconditional is written as

a

⇔

b

or

a

↔

b

with the latter being rare outside of

formal logic

. The

truth table

for the biconditional is

a

b

a

⇔

b

F

F

T

F

T

F

T

F

F

T

T

T

The biconditional

function

is often written as “iff,” meaning “if and only if.”

It gets its name from the fact that it is really two

conditionals

in

conjunction

,

(

a

→

b

)

∧

(

b

→

a

)

This fact is important to recognize when writing a mathematical proof, as both conditionals must be proven independently.

2

Colloquial Usage

The only unambiguous way of stating a biconditional in plain English is of the form “

b

if

a

and

a

if

b

.” Slightly more formal, one would say “

b

implies

a

and

a

implies

b

.” The plain English “if” may sometimes be used as a biconditional. One must weigh

context

heavily.

For example, “I’ll buy you an ice cream if you pass the exam” is meant as a biconditional, since the speaker doesn’t intend a valid outcome to be buying the ice cream whether or not you pass the exam (as in a conditional). However, “it is cloudy if it is raining” is

not

meant as a biconditional, since it can obviously be cloudy while not raining.

Title

biconditional

Canonical

name

Biconditional

Date of creation

2013-03-22 11:53:06

Last modified on

2013-03-22 11:53:06

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

17

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 03-00

Synonym

iff

Related topic

PropositionalLogic

Related topic

Equivalent3

Generated on Thu Feb 8 19:18:25 2018 by

LaTeXML
bijection

bijection

Let

X

and

Y

be sets. A

function

f

:

X

→

Y

that is

one-to-one

and onto is called a

bijection

or

bijective function

from

X

to

Y

.

When

X

=

Y

,

f

is also called a

permutation

of

X

.

An important

consequence

of the bijectivity of a function

f

is the

existence

of an

inverse function

f

-

1

. Specifically, a function is

invertible

if and only if it is

bijective

. Thus if

f

:

X

→

Y

is a bijection, then for any

A

⊂

X

and

B

⊂

Y

we have

f

∘

f

-

1

⁢

(

B

)

=

B

f

-

1

∘

f

⁢

(

A

)

=

A

It easy to see the

inverse

of a bijection is a bijection, and that a

composition

of bijections is again bijective.

Title

bijection

Canonical name

Bijection

Date of creation

2013-03-22 11:51:35

Last modified on

2013-03-22 11:51:35

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

16

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03-00

Classification

msc 83-00

Classification

msc 81-00

Classification

msc 82-00

Synonym

bijective

Synonym

bijective function

Synonym

1-1 correspondence

Synonym

1 to 1 correspondence

Synonym

one to one correspondence

Synonym

one-to-one correspondence

Related topic

Function

Related topic

Permutation

Related topic

InjectiveFunction

Related topic

Surjective

Related topic

Isomorphism2

Related topic

CardinalityOfAFiniteSetIsUnique

Related topic

CardinalityOfDisjointUnionOfFiniteSets

Related topic

AConnectedNormalSpaceWithMoreThanOnePointIsUncountable2

Related topic

AConnectedNormalSpaceWithMoreThanOnePointIsUncountable

Related topic

Bo

Generated on Thu Feb 8 19:18:27 2018 by

LaTeXML
Burali-Forti paradox

Burali-Forti paradox

The

Burali-Forti

paradox

demonstrates that the class of all

ordinals

is not a set. If there were a set of all ordinals,

O

⁢

r

⁢

d

, then it would follow that

O

⁢

r

⁢

d

was itself an ordinal, and therefore that

O

⁢

r

⁢

d

∈

O

⁢

r

⁢

d

. if sets in general are allowed to contain themselves, ordinals cannot since they are defined so that

∈

is well founded over them.

This paradox is

similar

to both Russell’s paradox and Cantor’s paradox, although it predates both. All of these paradoxes prove that a certain object is “too large” to be a set.

Title

Burali-Forti paradox

Canonical name

BuraliFortiParadox

Date of creation

2013-03-22 13:04:28

Last modified on

2013-03-22 13:04:28

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 03-00

Generated on Thu Feb 8 19:18:28 2018 by

LaTeXML
Cantor’s paradox

Cantor’s paradox

Cantor’s

paradox

demonstrates that there can be no largest

cardinality

. In particular, there must be an unlimited number of

infinite

cardinalities. For suppose that

α

were the largest

cardinal

. Then we would have

|

𝒫

⁢

(

α

)

|

=

|

α

|

. (Here

𝒫

⁢

(

α

)

denotes the

power set

of

α

.) Suppose

f

:

α

→

𝒫

⁢

(

α

)

is a

bijection

proving their equicardinality. Then

X

=

{

β

∈

α

∣

β

∉

f

⁢

(

β

)

}

is a subset of

α

, and so there is some

γ

∈

α

such that

f

⁢

(

γ

)

=

X

. But

γ

∈

X

↔

γ

∉

X

, which is a paradox.

The key part of the

argument

strongly resembles Russell’s paradox, which is in some sense a

generalization

of this paradox.

Besides allowing an

unbounded

number of cardinalities as ZF

set theory

does, this paradox could be avoided by a few other tricks, for

instance

by not allowing the construction of a power set or by adopting paraconsistent logic.

Title

Cantor’s paradox

Canonical name

CantorsParadox

Date of creation

2013-03-22 13:04:39

Last modified on

2013-03-22 13:04:39

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 03-00

Generated on Thu Feb 8 19:18:30 2018 by

LaTeXML
Cartesian product

Cartesian product

For any sets

A

and

B

, the

Cartesian product

A

×

B

is the set consisting of all

ordered pairs

(

a

,

b

)

where

a

∈

A

and

b

∈

B

.

The Cartesian product

satisfies

the following

properties

, for all sets

A

,

B

,

C

, and

D

:

•

A

×

∅

=

∅

•

(

A

×

B

)

∩

(

C

×

D

)

=

(

A

∩

C

)

×

(

B

∩

D

)

•

(

A

×

B

)

∁

=

(

A

∁

×

B

∁

)

∪

(

A

∁

×

B

)

∪

(

A

×

B

∁

)

Here

∅

denotes the

empty set

,

∩

denotes

intersection

,

∪

denotes union, and

∁

denotes

complement

with respect to some

universal set

U

containing

A

and

B

.

Title

Cartesian product

Canonical

name

CartesianProduct

Date of creation

2013-03-22 11:48:56

Last modified on

2013-03-22 11:48:56

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 03-00

Classification

msc 81P10

Classification

msc 81P05

Related topic

GeneralizedCartesianProduct

Generated on Thu Feb 8 19:18:32 2018 by

LaTeXML
characteristic function

characteristic function

Definition

Suppose

A

is a subset of a set

X

. Then the

function

χ

A

⁢

(

x

)

=

{

1

,

when

⁢

x

∈

A

,

0

,

when

⁢

x

∈

X

∖

A

is the

characteristic function

for

A

.

0.0.1

Properties

Suppose

A

,

B

are subsets of a set

X

.

1.

For set

intersections

and set unions, we have

χ

A

∩

B

=

χ

A

⁢

χ

B

,

χ

A

∪

B

=

χ

A

+

χ

B

-

χ

A

∩

B

,

χ

A

∩

B

=

min

⁡

(

χ

A

,

χ

B

)

,

χ

A

∪

B

=

max

⁡

(

χ

A

,

χ

B

)

.

2.

For the

symmetric difference

,

χ

A

△

B

=

χ

A

+

χ

B

-

2

⁢

χ

A

∩

B

.

3.

For the set

complement

,

χ

A

∁

=

1

-

χ

A

.

0.0.2

Remarks

A synonym for characteristic function is

indicator function

[

1

]

.

References

1

G.B. Folland,

Real Analysis

: Modern Techniques and Their Applications

, 2nd ed, John Wiley & Sons, Inc., 1999.

Title

characteristic function

Canonical name

CharacteristicFunction

Date of creation

2013-03-22 11:48:31

Last modified on

2013-03-22 11:48:31

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

12

Author

bbukh (348)

Entry type

Definition

Classification

msc 03-00

Classification

msc 26-00

Classification

msc 26A09

Classification

msc 28-00

Synonym

indicator function

Related topic

SimpleFunction

Generated on Thu Feb 8 19:18:38 2018 by

LaTeXML
characterization of signature of a permutation

characterization of signature of a permutation

The

signature

of a

permutation

is

well-defined

, as proved in the article. This note characterizes

odd permutations

.

Theorem 1

.

A permutation

σ

is odd if and only if the number of even-order cycles in its

cycle type

is odd.

Thus, for example, this

theorem

asserts that

(

1

⁢

2

⁢

3

)

is an

even permutation

, since it has zero even-order cycles, while

(

1

⁢

2

)

⁢

(

3

⁢

4

⁢

5

)

is odd, since it has precisely one even-order cycle.

Proof.

Note that the

function

taking a permutation to its signature is a

homomorphism

from

S

n

→

ℤ

/

2

⁢

ℤ

, and we thus get the following

multiplication

rules for even and odd permutations:

(even)

⋅

(even) = (odd)

⋅

(odd) = (even)

(even)

⋅

(odd) = (odd)

⋅

(even) = (odd)

Note that we can represent a single cycle as a

product

of

transpositions

:

(

a

1

⁢

a

2

⁢

a

3

⁢

…

⁢

a

k

)

=

(

a

1

⁢

a

k

)

⁢

(

a

1

⁢

a

k

-

1

)

⁢

…

⁢

(

a

1

⁢

2

)

and that therefore an even-length cycle is odd (since it is

equivalent

to an

odd number

of transpositions) while an odd-length cycle is even.

By the multiplication rules above, then, a given permutation is odd if and only if the product of the signs of its cycles is odd, which happens if and only if there are an odd number of cycles whose sign is odd, which happens if and only if there are an odd number of cycles of even length.
∎

Title

characterization

of signature of a permutation

Canonical

name

CharacterizationOfSignatureOfAPermutation

Date of creation

2013-03-22 17:16:49

Last modified on

2013-03-22 17:16:49

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 03-00

Classification

msc 05A05

Classification

msc 20B99

Generated on Thu Feb 8 19:18:40 2018 by

LaTeXML
concentric circles

concentric circles

A

collection

of circles is said to be

concentric

if they have the same center. The region formed between two

concentric circles

is therefore an annulus.

In the following picture, the gray region is the annulus delimited by the two black concentric circles.

Title

concentric circles

Canonical name

ConcentricCircles

Date of creation

2013-03-22 12:55:25

Last modified on

2013-03-22 12:55:25

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Definition

Classification

msc 03-00

Defines

annulus

Generated on Thu Feb 8 19:18:42 2018 by

LaTeXML
conjunction

conjunction

A

conjunction

is true only when both

parameters

(called conjuncts) are true. In English, conjunction is denoted by the word “and”. Symbolically, we represent it as

∧

or

multiplication

applied to

Boolean

parameters. Conjunction of

a

and

b

would be written

a

∧

b

or, in

algebraic

context

,

a

⋅

b

or

a

⁢

b

The

truth table

for conjunction is

a

b

a

∧

b

F

F

F

F

T

F

T

F

F

T

T

T

Title

conjunction

Canonical

name

Conjunction

Date of creation

2013-03-22 11:54:38

Last modified on

2013-03-22 11:54:38

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

15

Author

akrowne (2)

Entry type

Definition

Classification

msc 03-00

Synonym

logical and

Synonym

conjunctive truth function

Related topic

Disjunction

Related topic

PropositionalLogic

Generated on Thu Feb 8 19:18:44 2018 by

LaTeXML
criterion of surjectivity

criterion of surjectivity

Theorem

.

For surjectivity of a

mapping

f

:

A

→

B

,  it’s necessary and

sufficient

that

B

∖

f

⁢

(

X

)

⊆

f

⁢

(

A

∖

X

)

∀

X

⊆

A

.

(1)

Proof.

1

o

¯

.  Suppose that

f

:

A

→

B

is

surjective

.  Let

X

be an arbitrary subset of

A

and

y

any element of the set

B

∖

f

⁢

(

X

)

.  By the surjectivity, there is an

x

in

A

such that

f

⁢

(

x

)

=

y

, and since

y

∉

f

⁢

(

X

)

,  the

element

x

is not in

X

, i.e.

x

∈

A

∖

X

and thus

y

=

f

⁢

(

x

)

∈

f

⁢

(

A

∖

X

)

.  One can conclude that

B

∖

f

⁢

(

X

)

⊆

f

⁢

(

A

∖

X

)

for all

X

⊆

A

.

2

o

¯

.

Conversely

, suppose the condition (1).  Let again

X

be an arbitrary subset of

A

and

y

any element of

B

.  We have two possibilities:

a)

y

∉

f

⁢

(

X

)

; then

y

∈

B

∖

f

⁢

(

X

)

, and by (1),

y

∈

f

⁢

(

A

∖

X

)

.  This means that there exists an element

x

of

A

∖

X

⊆

A

such that

f

⁢

(

x

)

=

y

.

b)

y

∈

f

⁢

(

X

)

; then there exists an

x

∈

X

⊆

A

such that

f

⁢

(

x

)

=

y

.

The both cases show the surjectivity of

f

.

Title

criterion of surjectivity

Canonical name

CriterionOfSurjectivity

Date of creation

2013-03-22 18:04:56

Last modified on

2013-03-22 18:04:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Theorem

Classification

msc 03-00

Synonym

surjectivity criterion

Related topic

Function

Related topic

Image

Related topic

Subset

Generated on Thu Feb 8 19:18:47 2018 by

LaTeXML
cycle

cycle

Let

S

be a set. A

cycle

is a

permutation

(

bijective function

of a set onto itself)
such that there exist distinct

elements

a

1

,

a

2

,

…

,

a

k

of

S

such that

f

⁢

(

a

i

)

=

a

i

+

1

and

f

⁢

(

a

k

)

=

a

1

that is

f

⁢

(

a

1

)

=

a

2

f

⁢

(

a

2

)

=

a

3

⋮

f

⁢

(

a

k

)

=

a

1

and

f

⁢

(

x

)

=

x

for any other element of

S

.

This can also be pictured as

a

1

↦

a

2

↦

a

3

↦

⋯

↦

a

k

↦

a

1

and

x

↦

x

for any other element

x

∈

S

, where

↦

represents the action of

f

.

One of the basic results on

symmetric groups

says that any finite permutation can be expressed as

product

of

disjoint

cycles.

Title

cycle

Canonical

name

Cycle1

Date of creation

2013-03-22 12:24:23

Last modified on

2013-03-22 12:24:23

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Definition

Classification

msc 03-00

Classification

msc 05A05

Classification

msc 20F55

Related topic

Permutation

Related topic

SymmetricGroup

Related topic

Transposition

Related topic

Group

Related topic

Subgroup

Related topic

DihedralGroup

Related topic

CycleNotation

Related topic

PermutationNotation

Generated on Thu Feb 8 19:18:49 2018 by

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discontinuity of characteristic function

discontinuity of characteristic function

Theorem

.

For a subset

A

of

ℝ

n

, the set of the

discontinuity

(

http://planetmath.org/

Continuous

)
points of the

characteristic function

χ

A

is the
boundary (

http://planetmath.org/BoundaryFrontier

) of

A

.

Proof.

Let

a

be a discontinuity point of

χ

A

.  Then any

neighborhood

(

http://planetmath.org/Neighborhood

) of

a

contains the points

b

and

c

such that

χ

A

⁢

(

b

)

=

1

and

χ

A

⁢

(

c

)

=

0

.  Thus

b

∈

A

and

c

∉

A

,  whence

a

is a boundary point of

A

.

If, on the contrary,

a

is a boundary point of

A

and

U

⁢

(

a

)

an arbitrary neighborhood of

a

, it follows that

U

⁢

(

a

)

contains both points belonging to

A

and points not belonging to

A

.  So we have in

U

⁢

(

a

)

the points

b

and

c

such that

χ

A

⁢

(

b

)

=

1

and

χ

A

⁢

(

c

)

=

0

.  This means that

χ

A

cannot be continuous at the point

a

(N.B. that
one does not need to know the value

χ

A

⁢

(

a

)

).

Title

discontinuity of characteristic function

Canonical name

DiscontinuityOfCharacteristicFunction

Date of creation

2015-02-03 21:23:33

Last modified on

2015-02-03 21:23:33

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

3

Author

pahio (2872)

Entry type

Theorem

Classification

msc 03-00

Classification

msc 26-00

Classification

msc 26A09

Generated on Thu Feb 8 19:18:51 2018 by

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disjoint

disjoint

Two sets

X

and

Y

are

disjoint

if their

intersection

X

∩

Y

is the

empty set

.

Title

disjoint

Canonical

name

Disjoint

Date of creation

2013-03-22 11:55:14

Last modified on

2013-03-22 11:55:14

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 03-00

Generated on Thu Feb 8 19:18:53 2018 by

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empty set

empty set

An

empty set

is a set

∅

that contains no

elements

. The

Zermelo-Fraenkel Axioms

of

set theory

imply that there exists an empty set. One constructs an empty set by starting with any set

X

and then applying the

axiom of separation

to form the empty set

∅

:=

{

x

∈

X

∣

x

≠

x

}

.

An empty set is a subset of every other set, and any two empty sets are equal.

Alternative notations

for the empty set include

{

}

and

∅

.

Title

empty set

Canonical name

EmptySet

Date of creation

2013-03-22 11:49:55

Last modified on

2013-03-22 11:49:55

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 03-00

Classification

msc 65H05

Classification

msc 65H10

Synonym

null set

Generated on Thu Feb 8 19:18:54 2018 by

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example of an operator

example of an operator

While the

operator

and

function

are often used as synonyms, it is customary in some of mathematics to use the operator in of

mapping

, function or

transformation

. In particular, in

linear algebra

a transformation

T

:

V

→

W

is referred to as an operator on

V

if

W

=

V

. In general, in the of

Algebra

and

Analysis

the operator is preferred for transformations whose codomain is a subset of its domain.

Title

example of an operator

Canonical name

ExampleOfAnOperator

Date of creation

2013-03-22 15:50:04

Last modified on

2013-03-22 15:50:04

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Example

Classification

msc 03-00

Generated on Thu Feb 8 19:18:59 2018 by

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example of bijection

example of bijection

It is known that the

Cartesian product

of numerable sets are numerable, then the

product

ℕ

×

ℕ

is. Here we give an explicit

mapping

which does the task:

(

n

,

m

)

↦

m

+

(

n

+

m

)

⁢

(

n

+

m

+

1

)

2

It is almost easy to check bijectivity and curious how the elements of

ℕ

×

ℕ

are labeled.

Title

example of

bijection

Canonical name

ExampleOfBijection

Date of creation

2013-03-22 16:30:18

Last modified on

2013-03-22 16:30:18

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

7

Author

juanman (12619)

Entry type

Example

Classification

msc 03-00

Related topic

ParingFunction

Generated on Thu Feb 8 19:19:01 2018 by

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fix

fix

In mathematical statements, mathematical objects such as points and numbers are described as being

fixed

. A possible meaning for this usage is that the mathematical object in question is not allowed to vary throughout the statement or proof (or, in some cases, a portion thereof). Although a fixed object typically does not vary, it is almost always arbitrary. This may seem

paradoxical

, but it is quite logical: An object is chosen arbitrarily, then it is never allowed to vary. See the entry betweenness in rays for an example of this usage.

The usage of the

fix

and

fixed

may also that a

mapping

sends the mathematical object to itself. These two usages are technically not the same. The former usage (described in the previous paragraph) states a

property

of the mathematical object in question and is always either part of an

implication

(as in “If

x

∈

ℝ

is fixed, then…”) or a command made by the author to the reader (as in “Let

x

∈

ℝ

be fixed.” and “Fix

x

∈

ℝ

.”). The latter usage (described in this paragraph) states a property of a mapping and may or may not be part of a

conditional

statement or a command. The word “fixes”

always

refers to this usage (as in “Note that

f

fixes

x

.”). See the entry fix (

transformation

actions) (

http://planetmath.org/Fixed

) for a further explanation of the latter usage.

Title

fix

Canonical

name

Fix

Date of creation

2013-03-22 16:11:19

Last modified on

2013-03-22 16:11:19

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03-00

Classification

msc 03F07

Synonym

fixed

Related topic

Fixed

Generated on Thu Feb 8 19:19:04 2018 by

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fixed point

fixed point

A

fixed point

x

of a

function

f

:

X

→

X

, is a point that remains

constant

upon application of that function, i.e.:

f

⁢

(

x

)

=

x

.

Title

fixed point

Canonical name

FixedPoint

Date of creation

2013-03-22 12:32:05

Last modified on

2013-03-22 12:32:05

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 03-00

Related topic

BanachFixedPointTheorem

Related topic

BrouwerFixedPointTheorem

Related topic

SchauderFixedPointTheorem

Related topic

FixedPointProperty

Related topic

TarskiKnasterTheorem

Generated on Thu Feb 8 19:19:05 2018 by

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