aarjaneiro/planetmath huggingface dataset:
injective function

injective function

We say that a

function

f

:

A

→

B

is

injective

or

one-to-one

if

f

⁢

(

x

)

=

f

⁢

(

y

)

implies

x

=

y

, or equivalently, whenever

x

≠

y

, then

f

⁢

(

x

)

≠

f

⁢

(

y

)

.

Properties

1.

Suppose

A

,

B

,

C

are sets and

f

:

A

→

B

,

g

:

B

→

C

are injective functions. Then the

composition

g

∘

f

is an injection.

2.

Suppose

f

:

A

→

B

is an injection, and

C

⊆

A

. Then
the

restriction

f

|

C

:

C

→

B

is an injection.

For a list of other of
injective functions, see

[

1

]

.

References

1

Wikipedia, article on

http://en.wikipedia.org/wiki/Injective_function

Injective function.

Title

injective function

Canonical

name

InjectiveFunction

Date of creation

2013-03-22 11:51:38

Last modified on

2013-03-22 11:51:38

Owner

drini (3)

Last modified by

drini (3)

Numerical id

16

Author

drini (3)

Entry type

Definition

Classification

msc 03E20

Classification

msc 03E99

Synonym

one-to-one

Synonym

injection

Synonym

embedding

Synonym

injective

Related topic

Bijection

Related topic

Function

Related topic

Surjective

Generated on Thu Feb 8 20:13:08 2018 by

LaTeXML
invariant

invariant

Let

A

be a set, and

T

:

A

→

A

a

transformation

of that
set. We say that

x

∈

A

is

an

invariant

of

T

whenever

x

is
fixed by

T

:

T

⁢

(

x

)

=

x

.

We say that a subset

B

⊂

A

is

invariant with respect to

T

whenever

T

⁢

(

B

)

⊂

B

.

If this is so, the

restriction

of

T

is a

well-defined

transformation of the

invariant subset

:

T

|

B

:

B

→

B

.

The

definition

generalizes readily to a family of transformations with
common domain

T

i

:

A

→

A

,

i

∈

I

In this case we say that a subset is invariant, if it is invariant
with respect to all elements of the family.

Title

invariant

Canonical

name

Invariant

Date of creation

2013-03-22 12:26:09

Last modified on

2013-03-22 12:26:09

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

8

Author

rmilson (146)

Entry type

Definition

Classification

msc 03E20

Related topic

Transformation

Related topic

InvariantSubspace

Related topic

Fixed

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

LaTeXML
invariant by a measure-preserving transformation

invariant by a measure-preserving transformation

\PMlinkescapephrase

invariant

\PMlinkescapephrase

properties

\PMlinkescapephrase

property

Let

X

be a set and

T

:

X

⟶

X

a

transformation

of

X

.

The notion of invariance by

T

we are about to describe is stronger than the usual notion of invariance (

http://planetmath.org/invariant

), and is especially useful in

ergodic theory

. Thus, in most applications,

(

X

,

𝔅

,

μ

)

is a

measure space

and

T

is a

measure-preserving transformation

. Nevertheless, the definition of invariance and its properties are general and do not require any such assumptions.

Definition -

A subset

A

⊆

X

is said to be

invariant

by

T

, or

T

-

invariant

, if

T

-

1

⁢

(

A

)

=

A

.

The fundamental property of this

concept

is the following: if

A

is invariant by

T

, then so is

X

∖

A

.

Thus, when

A

is invariant by

T

we obtain by

restriction

two well-defined transformations

T

|

A

:

A

⟶

A

T

|

X

∖

A

:

X

∖

A

⟶

X

∖

A

Hence, the existence of an allows one to decompose the set

X

into two disjoint subsets and study the transformation

T

in each of these subsets.

Remark -

When

T

is a measure-preserving transformation in a measure space

(

X

,

𝔅

,

μ

)

one usually restricts the notion of invariance to measurable subsets

A

∈

𝔅

.

Title

invariant by a measure-preserving transformation

Canonical name

InvariantByAMeasurepreservingTransformation

Date of creation

2013-03-22 18:04:15

Last modified on

2013-03-22 18:04:15

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Definition

Classification

msc 03E20

Classification

msc 28D05

Classification

msc 37A05

Generated on Thu Feb 8 20:13:11 2018 by

LaTeXML
inverse function theorem

inverse function theorem

Let

𝐟

be a

continuously differentiable

,

vector-valued function

mapping the open set

E

⊂

ℝ

n

to

ℝ

n

and let

S

=

𝐟

⁢

(

E

)

. If, for some point

𝐚

∈

E

, the

Jacobian

,

|

J

𝐟

⁢

(

𝐚

)

|

, is non-zero, then there is a uniquely defined

function

𝐠

and two open sets

X

⊂

E

and

Y

⊂

S

such that

1.

𝐚

∈

X

,

𝐟

⁢

(

𝐚

)

∈

Y

;

2.

Y

=

𝐟

⁢

(

X

)

;

3.

𝐟

:

X

→

Y

is one-one;

4.

𝐠

is continuously differentiable on

Y

and

𝐠

⁢

(

𝐟

⁢

(

𝐱

)

)

=

𝐱

for all

𝐱

∈

X

.

0.0.1

Simplest case

When

n

=

1

, this theorem becomes: Let

f

be a continuously differentiable, real-valued function defined on the

open interval

I

. If for some point

a

∈

I

,

f

′

⁢

(

a

)

≠

0

, then there is a neighbourhood

[

α

,

β

]

of

a

in which

f

is

strictly monotonic

. Then

y

→

f

-

1

⁢

(

y

)

is a continuously differentiable, strictly monotonic function from

[

f

⁢

(

α

)

,

f

⁢

(

β

)

]

to

[

α

,

β

]

. If

f

is increasing (or decreasing) on

[

α

,

β

]

, then so is

f

-

1

on

[

f

⁢

(

α

)

,

f

⁢

(

β

)

]

.

0.0.2

Note

The

inverse function theorem

is a special case of the

implicit function theorem

where the

dimension

of each variable is the same.

Title

inverse function theorem

Canonical name

InverseFunctionTheorem

Date of creation

2013-03-22 12:58:30

Last modified on

2013-03-22 12:58:30

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

9

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 03E20

Related topic

DerivativeOfInverseFunction

Related topic

LegendreTransform

Related topic

DerivativeAsParameterForSolvingDifferentialEquations

Related topic

TheoryForSeparationOfVariables

Generated on Thu Feb 8 20:13:14 2018 by

LaTeXML
inverse image

inverse image

Let

f

:

A

⟶

B

be a

function

, and let

U

⊂

B

be a subset. The

inverse image

of

U

is the set

f

-

1

⁢

(

U

)

⊂

A

consisting of all

elements

a

∈

A

such that

f

⁢

(

a

)

∈

U

.

The inverse image commutes with all set

operations

: For any

collection

{

U

i

}

i

∈

I

of subsets of

B

, we have the following

identities

for

1.

Unions:

f

-

1

⁢

(

⋃

i

∈

I

U

i

)

=

⋃

i

∈

I

f

-

1

⁢

(

U

i

)

2.

Intersections

:

f

-

1

⁢

(

⋂

i

∈

I

U

i

)

=

⋂

i

∈

I

f

-

1

⁢

(

U

i

)

and for any subsets

U

and

V

of

B

, we have identities for

3.

Complements

:

(

f

-

1

⁢

(

U

)

)

∁

=

f

-

1

⁢

(

U

∁

)

4.

Set differences

:

f

-

1

⁢

(

U

∖

V

)

=

f

-

1

⁢

(

U

)

∖

f

-

1

⁢

(

V

)

5.

Symmetric differences

:

f

-

1

⁢

(

U

△

V

)

=

f

-

1

⁢

(

U

)

△

f

-

1

⁢

(

V

)

In

addition

, for

X

⊂

A

and

Y

⊂

B

, the inverse image

satisfies

the miscellaneous identities

6.

(

f

|

X

)

-

1

⁢

(

Y

)

=

X

∩

f

-

1

⁢

(

Y

)

7.

f

⁢

(

f

-

1

⁢

(

Y

)

)

=

Y

∩

f

⁢

(

A

)

8.

X

⊂

f

-

1

⁢

(

f

⁢

(

X

)

)

, with equality if

f

is

injective

.

Title

inverse image

Canonical name

InverseImage

Date of creation

2013-03-22 11:51:58

Last modified on

2013-03-22 11:51:58

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 03E20

Classification

msc 46L05

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Synonym

preimage

Related topic

Mapping

Related topic

DirectImage

Generated on Thu Feb 8 20:13:16 2018 by

LaTeXML
irreflexive

irreflexive

A

binary relation

ℛ

on a set

A

is said to be

irreflexive

(or

antireflexive

) if

∀

a

∈

A

,

¬

⁢

a

⁢

ℛ

⁢

a

. In other words, “no

element

is

ℛ

-related to itself.”

For example, the

relation

<

(“less than”) is an irreflexive relation on the set of natural numbers.

Note that “irreflexive” is not simply the

negation

of “

reflexive

(

http://planetmath.org/Reflexive

)
.” Although it is impossible for a relation (on a

nonempty set

) to be both reflexive (

http://planetmath.org/Reflexive

)
and irreflexive, there exist relations that are neither. For example, the relation

{

(

a

,

a

)

}

on the two element set

{

a

,

b

}

is neither reflexive nor irreflexive.

Here is an example of a non-reflexive, non-irreflexive relation “in nature.” A

subgroup

in a group is said to be

self-normalizing

if it is equal to its own

normalizer

. For a group

G

, define a relation

ℛ

on the set of all subgroups of

G

by declaring

H

⁢

ℛ

⁢

K

if and only if

H

is the normalizer of

K

. Notice that every nontrivial group has a subgroup that is not self-normalizing; namely, the trivial subgroup

{

e

}

consisting of only the

identity

. Thus, in any nontrivial group

G

, there is a subgroup

H

of

G

such that

¬

⁢

H

⁢

ℛ

⁢

H

. So the relation

ℛ

is non-reflexive. Moreover, since the normalizer of a group

G

in

G

is

G

itself, we have

G

⁢

ℛ

⁢

G

. So

ℛ

is non-irreflexive.

Title

irreflexive

Canonical name

Irreflexive

Date of creation

2013-03-22 15:41:45

Last modified on

2013-03-22 15:41:45

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E20

Synonym

antireflexive

Related topic

Reflexive

Generated on Thu Feb 8 20:13:18 2018 by

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left function notation

left function notation

We are said to be using
if we write

functions

to the left of their

arguments

.
That is, if

α

:

X

→

Y

is a function and

x

∈

X

,
then

α

⁢

x

is the image of

x

under

α

.

Furthermore, if we have a function

β

:

Y

→

Z

,
then we write the

composition

of the two functions
as

β

⁢

α

:

X

→

Z

,
and the image of

x

under the composition
as

β

⁢

α

⁢

x

=

(

β

⁢

α

)

⁢

x

=

β

⁢

(

α

⁢

x

)

.

Compare this to

right function notation

.

Title

left function notation

Canonical name

LeftFunctionNotation

Date of creation

2013-03-22 12:09:18

Last modified on

2013-03-22 12:09:18

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 03E20

Synonym

left notation

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

LaTeXML
level set

level set

Let

A

and

B

be sets and

f

a function mapping

A

into

B

. Then the sets

{

x

∣

f

⁢

(

x

)

=

y

}

where

y

∈

B

are known as the

level sets

(or

fibres

) of

f

.

Title

level set

Canonical

name

LevelSet

Date of creation

2013-03-22 14:53:20

Last modified on

2013-03-22 14:53:20

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

6

Author

yark (2760)

Entry type

Definition

Classification

msc 03E20

Related topic

Fibre

Related topic

ContourLine

Generated on Thu Feb 8 20:13:22 2018 by

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level sets of smooth functions on manifolds

level sets of smooth functions on manifolds

Let

f

:

ℝ

n

→

ℝ

be smooth. Further suppose that the

gradient

of

f

differs from zero at every point of a

level set

. Then it follows from the

implicit function theorem

that that level set is a smooth hypersurface. Furthermore, at any point of the level set, the gradient of the

function

at that point is

orthogonal

to the level set.

One can generalize this observation to manifolds. Suppose that

M

is a smooth manifold and that

f

:

M

→

ℝ

is smooth. Further suppose that the gradient of

f

differs from zero at every point of a level set. Then it follows from the implicit function theorem that that level set is a smooth hypersurface. If one chooses a Riemannian metric on the manifold, the gradient of the function at that point will be orthogonal to the level set.

Title

level sets of smooth functions on manifolds

Canonical

name

LevelSetsOfSmoothFunctionsOnManifolds

Date of creation

2013-03-22 15:20:02

Last modified on

2013-03-22 15:20:02

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03E20

Generated on Thu Feb 8 20:13:23 2018 by

LaTeXML
mapping

mapping

The term

mapping

is a synonym of

function

(

http://planetmath.org/Function

), although usage

patterns

suggest that “mapping” is the more

generic

term.

In a geometric

context

, the term “function” often connotes a mapping whose purpose is to assign values to the elements of its domain. In other words, a function defines a field of values. By contrast, “mapping” has a more geometric

connotation

, as in “a mapping of one space to another”.

Title

mapping

Canonical name

Mapping

Date of creation

2013-03-22 12:20:16

Last modified on

2013-03-22 12:20:16

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

18

Author

rmilson (146)

Entry type

Definition

Classification

msc 03E20

Synonym

map

Related topic

DirectImage

Related topic

InverseImage

Related topic

Domain

Related topic

Codomain

Related topic

Set

Related topic

Function

Related topic

Transformation

Generated on Thu Feb 8 20:13:29 2018 by

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mapping of period n is a bijection

mapping of period

n

is a bijection

Theorem

Suppose

X

is a set.
Then a

mapping

f

:

X

→

X

of period (

http://planetmath.org/PeriodOfMapping

)

n

is
a

bijection

.

Proof.

If

n

=

1

, the claim is

trivial

;

f

is the

identity mapping

.
Suppose

n

=

2

,

3

,

…

.
Then for any

x

∈

X

, we have

x

=

f

⁢

(

f

n

-

1

⁢

(

x

)

)

,
so

f

is an

surjection

. To see that

f

is a

injection

,
suppose

f

⁢

(

x

)

=

f

⁢

(

y

)

for some

x

,

y

in

X

. Since

f

n

is the

identity

, it follows that

x

=

y

.

□

Title

mapping of period

n

is a bijection

Canonical name

MappingOfPeriodNIsABijection

Date of creation

2013-03-22 13:48:57

Last modified on

2013-03-22 13:48:57

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Proof

Classification

msc 03E20

Generated on Thu Feb 8 20:13:31 2018 by

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multigrade operator

multigrade operator

A

multigrade operator

Ω

is a

parametric operator

with parameter

k

in the set

ℕ

of non-negative integers.

The application of a multigrade operator

Ω

to a

finite sequence

of operands

(

x

1

,

…

,

x

k

)

is typically denoted with the parameter

k

left tacit, as the appropriate application is implicit in the number of operands listed. Thus

Ω

⁢

(

x

1

,

…

,

x

k

)

may be taken for

Ω

k

⁢

(

x

1

,

…

,

x

k

)

.

Title

multigrade operator

Canonical name

MultigradeOperator

Date of creation

2013-03-22 17:48:11

Last modified on

2013-03-22 17:48:11

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

4

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 03E20

Classification

msc 03C05

Classification

msc 08A40

Classification

msc 08A70

Related topic

ParametricOperator

Generated on Thu Feb 8 20:13:32 2018 by

LaTeXML
parametric operator

parametric operator

A

parametric operator

Ω

with

parameter

α

in the

parametric set

𝒜

is an indexed family of

operators

(

Ω

α

)

𝒜

=

{

Ω

α

:

α

∈

𝒜

}

with index

α

in the

index set

𝒜

.

Title

parametric operator

Canonical

name

ParametricOperator

Date of creation

2013-03-22 17:48:04

Last modified on

2013-03-22 17:48:04

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

6

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 03E20

Classification

msc 03C05

Related topic

MultigradeOperator

Generated on Thu Feb 8 20:13:49 2018 by

LaTeXML
partial function

partial function

A

function

f

:

A

→

B

is sometimes called a

total function

, to signify that

f

⁢

(

a

)

is defined for every

a

∈

A

. If

C

is any set such that

C

⊇

A

then

f

is also a

partial function

from

C

to

B

.

Clearly if

f

is a function from

A

to

B

then it is a partial function from

A

to

B

, but a partial function need not be defined for every element of its domain. The set of elements of

A

for which

f

is defined is sometimes called the

domain of definition

.

Title

partial function

Canonical name

PartialFunction

Date of creation

2013-03-22 12:58:15

Last modified on

2013-03-22 12:58:15

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

11

Author

Henry (455)

Entry type

Definition

Classification

msc 03E20

Defines

total function

Defines

domain of definition

Generated on Thu Feb 8 20:13:50 2018 by

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partial mapping

partial mapping

Let

X

1

,

⋯

,

X

n

and

Y

be sets, and let

f

be a

function

of

n

variables

:

f

:

X

1

×

X

2

×

⋯

×

X

n

→

Y

.

x

i

∈

X

i

for

2

≤

i

≤

n

. The induced

mapping

a

↦

f

⁢

(

a

,

x

2

,

…

,

x

n

)

is called the

partial mapping

determined by

f

corresponding to the first variable.

In the case where

n

=

2

, the map defined by

a

↦

f

⁢

(

a

,

x

)

is often denoted

f

⁢

(

⋅

,

x

)

. Further, any function

f

:

X

1

×

X

2

→

Y

determines a mapping
from

X

1

into the set of mappings of

X

2

into

Y

, namely

f

¯

:

x

↦

(

y

↦

f

(

x

,

y

)

)

.
The

converse

holds too, and it is customary to identify

f

with

f

¯

. Many of the “

canonical isomorphisms

” that we come across (e.g. in multilinear algebra) are illustrations of this kind of identification.

Title

partial mapping

Canonical

name

PartialMapping

Date of creation

2013-03-22 13:59:31

Last modified on

2013-03-22 13:59:31

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03E20

Generated on Thu Feb 8 20:13:52 2018 by

LaTeXML
period of mapping

period of mapping

Definition

Suppose

X

is a set and

f

is a

mapping

f

:

X

→

X

.
If

f

n

is the

identity mapping

on

X

for some

n

=

1

,

2

,

…

, then

f

is said to be a

mapping of period

n

.
Here, the notation

f

n

means the

n

-fold

composition

f

∘

⋯

∘

f

.

0.0.1

Examples

1.

A mapping

f

is of period

1

if and only if

f

is the identity
mapping.

2.

Suppose

V

is a

vector space

. Then a

linear involution

L

:

V

→

V

is
a mapping of period

2

.
For example, the

reflection

mapping

x

↦

-

x

is a mapping of period

2

.

3.

In the complex plane, the mapping

z

↦

e

-

2

⁢

π

⁢

i

/

n

⁢

z

is a
mapping of period

n

for

n

=

1

,

2

,

…

.

4.

Let us consider the

function space

spanned by the

trigonometric functions

sin

and

cos

. On this space,
the

derivative

is a mapping of period

4

.

0.0.2

Properties

1.

Suppose

X

is a set.
Then a mapping

f

:

X

→

X

of period

n

is a

bijection

.
(proof.) (

http://

planetmath

.org/MappingOfDegreeNIsASurjection

)

2.

Suppose

X

is a

topological space

.
Then a

continuous mapping

f

:

X

→

X

of period

n

is a

homeomorphism

.

Title

period of mapping

Canonical name

PeriodOfMapping

Date of creation

2013-03-22 13:48:53

Last modified on

2013-03-22 13:48:53

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

12

Author

bwebste (988)

Entry type

Definition

Classification

msc 03E20

Related topic

Retract

Related topic

Idempotency

Generated on Thu Feb 8 20:13:54 2018 by

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pi-system

pi-system

Let

Ω

be a set, and

𝒫

⁢

(

Ω

)

be the

power set

of

Ω

. A

π

-system

(or

pi-system

) on

Ω

is a set

ℱ

⊆

𝒫

⁢

(

Ω

)

such that

A

,

B

∈

ℱ

⇒

A

∩

B

∈

ℱ

.

(1)

A

π

-system is

closed under

finite

intersection

.

Title

pi-system

Canonical

name

Pisystem

Date of creation

2013-03-22 12:21:23

Last modified on

2013-03-22 12:21:23

Owner

drummond (72)

Last modified by

drummond (72)

Numerical id

4

Author

drummond (72)

Entry type

Definition

Classification

msc 03E20

Classification

msc 28A60

Related topic

DynkinSystem

Related topic

SigmaAlgebra

Related topic

DynkinsLemma

Generated on Thu Feb 8 20:13:56 2018 by

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product

product

The word

product

in mathematics generally means the result of some of

multiplication

operation

, i.e. (

http://planetmath.org/Ie

) of certain of

mapping

X

×

X

→

Y

; such operations are commonly

distributive

over the

addition

operation of

X

if it is defined.

If

x

1

and

x

2

are two elements of the set

X

, giving the product

y

∈

Y

, then

x

1

and

x

2

are in general called the

factors

of this product.

Some most usual products are

•

the

ring product

(

http://planetmath.org/Ring

) (also in fields), especially the product of numbers and the product of

square matrices

;

•

on vectors the

scalar product

, the

vector product

, the

dyad product

and the

Hadamard product

; on ideals the

product of ideals

;

•

the

Cartesian product

, the

direct products

of various systems (not in with any additions).

Such kinds of product that are

associative

, allow to form a product of more than two factors, which is justified by the

theorem

in the entry

general associativity

.  E.g. the usual product of the integers from 1 to

n

is the factorial of

n

.

Title

product

Canonical name

Product

Date of creation

2013-03-22 15:45:22

Last modified on

2013-03-22 15:45:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 03E20

Related topic

Multiplication

Related topic

RuleOfProduct

Related topic

FrobeniusProduct

Defines

factor

Defines

factor of a product

Generated on Thu Feb 8 20:14:00 2018 by

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product map

product map

Notation:

If

{

X

i

}

i

∈

I

is a

collection

of sets (

indexed by

I

) then

∏

i

∈

I

X

i

denotes the

generalized Cartesian product

of

{

X

i

}

ı

∈

I

.

Let

{

A

i

}

i

∈

I

and

{

B

i

}

i

∈

I

be collections of sets indexed by the same set

I

and

f

i

:

A

i

⟶

B

i

a collection of

functions

.

The

product map

is the function

∏

i

∈

I

f

i

:

∏

i

∈

I

A

i

⟶

∏

i

∈

I

B

i

(

∏

i

∈

I

f

i

)

⁢

(

a

i

)

i

∈

I

:=

(

f

i

⁢

(

a

i

)

)

i

∈

I

0.1

Properties:

•

If

f

i

:

A

i

⟶

B

i

and

g

i

:

B

i

⟶

C

i

are collections of functions then

∏

i

∈

I

g

i

∘

∏

i

∈

I

f

i

=

∏

i

∈

I

g

i

∘

f

i

•

∏

i

∈

I

f

i

is

injective

if and only if each

f

i

is injective.

•

∏

i

∈

I

f

i

is

surjective

if and only if each

f

i

is surjective.

•

Suppose

{

A

i

}

i

∈

I

and

{

B

i

}

i

∈

I

are

topological spaces

. Then

∏

i

∈

I

f

i

is

continuous

(

http://planetmath.org/Continuous

) (in the

product topology

) if and only if each

f

i

is continuous.

•

Suppose

{

A

i

}

i

∈

I

and

{

B

i

}

i

∈

I

are groups, or rings or

algebras

. Then

∏

i

∈

I

f

i

is a group (ring or )

homomorphism

if and only if each

f

i

is a group (ring or ) homomorphism.

Title

product map

Canonical name

ProductMap

Date of creation

2013-03-22 17:48:28

Last modified on

2013-03-22 17:48:28

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Definition

Classification

msc 03E20

Generated on Thu Feb 8 20:14:03 2018 by

LaTeXML