aarjaneiro/planetmath huggingface dataset:
Formulas for Students of Class 8th.

Formulas for Students of Class 8th.

1. Area of

Trapezium

-¿ 1/2*Sum of

parallel

sides*

Distance

between parallel sides.

2. Area of

Rectangle

-¿ Length*Breadth

3.

Perimeter

of Rectangle -¿ 2(Length+Breadth)

4. Area of Square -¿ Side*Side

5. Perimeter of Square -¿ 4*Side

6. Area of Genereal

Quadrilateral

-¿ 1/2*d(h1+h2) ,where ”d” stands for

diameter

and ”h1” and ”h2” stands for height first and height sceond.

7.

Area of Triangle

-¿ 1/2*breadth*height

8. Perimeter of

Triangle

-¿ Sum of all 3 sides.

9.

Area of Quadrilateral

-¿ 4*Side*Side

10. Area of

Rhombus

-¿ 1/2*D1*D2 , where ”D1 and D2” stands for Diameter first and Diameter second.

I will post some more formulas later.

–Mrigendra

Title

Formulas for Students of Class 8th.

Canonical name

FormulasForStudentsOfClass8th

Date of creation

2013-03-11 19:14:42

Last modified on

2013-03-11 19:14:42

Owner

Expert (27887)

Last modified by

(0)

Numerical id

4

Author

Expert (0)

Entry type

Theorem

Classification

msc 03-00

Synonym

Related topic

Defines

Generated on Thu Feb 8 19:19:07 2018 by

LaTeXML
functions from empty set

functions from empty set

Sometimes, it is useful to consider

functions

whose domain is the

empty set

. Given a set, there exists exactly one function from
the the empty set to that set. The rationale for this comes from
carefully examining the

definition

of function in this degenerate
case. Recall that, in

set theory

, a function from a set

D

to a
set

R

is a set of

ordered pairs

whose first

element

lies in

D

and whose second element lies in

R

such that every element of

D

appears as the first element of exactly one ordered pair. If we
take

D

to be the empty set, we see that this definition is satisfied
if we take our function to be set of no ordered pairs — since there
are no elements in the empty set, it is technically correct to say
that every element of the empty set appears as a first element of
an ordered pair which is an element of the empty set!

This observation turns out to be more than just an exercise in
logic, being useful in several

contexts

. Given a set

S

and a positive
integer

n

, we may define

S

n

as the set of all functions from

{

1

,

…

,

n

}

to

S

. If we choose

n

=

0

, then

S

0

consists
of all maps from the empty set to

S

, hence consists of exactly one
element — see the entry on

empty products

for a discussion of the
usefulness of this convention. In

category theory

, it turns out that
functions from the empty set are important because they make the empty
set be an

initial object

in this

category

.

Title

functions from empty set

Canonical

name

FunctionsFromEmptySet

Date of creation

2013-03-22 18:08:20

Last modified on

2013-03-22 18:08:20

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03-00

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

LaTeXML
inverse function

inverse function

Definition

Suppose

f

:

X

→

Y

is a

function

between sets

X

and

Y

,
and suppose

f

-

1

:

Y

→

X

is a

mapping

that

satisfies

f

-

1

∘

f

=

id

X

,

f

∘

f

-

1

=

id

Y

,

where

id

A

denotes the

identity function

on the set

A

.
Then

f

-

1

is called the

inverse

of

f

,
or the

inverse function

of

f

.
If

f

has an inverse near a point

x

∈

X

, then

f

is

invertible

near

x

. (That is, if there is a set

U

containing

x

such that the

restriction

of

f

to

U

is invertible, then

f

is invertible
near

x

.) If

f

is invertible near all

x

∈

X

, then

f

is

invertible

.

Properties

1.

When an inverse function exists, it is unique.

2.

The inverse function and the

inverse image

of a set coincide
in the following sense.
Suppose

f

-

1

⁢

(

A

)

is the inverse image of a set

A

⊂

Y

under a function

f

:

X

→

Y

.
If

f

is a

bijection

, then

f

-

1

⁢

(

y

)

=

f

-

1

⁢

(

{

y

}

)

.

3.

The inverse function of a function

f

:

X

→

Y

exists if and only
if

f

is a bijection, that is,

f

is an

injection

and a

surjection

.

4.

A

linear mapping

between

vector spaces

is invertible if and only if
the

determinant

of the mapping is nonzero.

5.

For

differentiable functions

between

Euclidean spaces

, the

inverse function
theorem

gives a necessary and

sufficient

condition for the inverse to exist.
This can be generalized to maps between Banach spaces which are

differentiable

in the sense of Frechet.

Remarks

When

f

is a linear mapping (for

instance

, a matrix), the term

non-singular

is
also used as a synonym for invertible.

Title

inverse function

Canonical name

InverseFunction

Date of creation

2013-03-22 13:53:52

Last modified on

2013-03-22 13:53:52

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

14

Author

matte (1858)

Entry type

Definition

Classification

msc 03-00

Classification

msc 03E20

Synonym

non-singular function

Synonym

nonsingular function

Synonym

non-singular

Synonym

nonsingular

Synonym

inverse

Related topic

Function

Defines

invertible function

Defines

invertible

Generated on Thu Feb 8 19:19:11 2018 by

LaTeXML
inverse of composition of functions

inverse of composition of functions

Theorem

.

Let

f

and

g

be

invertible functions

such that their

composition

f

∘

g

is well defined. Then

f

∘

g

is invertible and

(

f

∘

g

)

-

1

=

g

-

1

∘

f

-

1

.

Before proving this

theorem

, it should be noted that some students encounter this result long before they are introduced to formal proof. Fortunately, there is an intuitive way to think about this theorem: Think of the function

g

as putting on one’s socks and the function

f

as putting on one’s shoes. Then

f

∘

g

denotes the process of putting one one’s socks, then putting on one’s shoes. (Recall that function composition works from right to left.) Note that

(

f

∘

g

)

-

1

refers to the reverse process of

f

∘

g

, which is taking off one’s shoes (which is

f

-

1

) followed by taking off one’s socks (which is

g

-

1

).

Due to the intuitive

argument

given above, the theorem is referred to as the

socks and shoes rule

. This name is a

mnemonic device

which reminds people that, in order to obtain the

inverse

of a composition of functions, the original functions have to be undone in the opposite order.

Now for the formal proof.

Proof.

Let

A

,

B

, and

C

be sets such that

g

:

A

→

B

and

f

:

B

→

C

. Then the following two equations must be shown to hold:

(

g

-

1

∘

f

-

1

)

∘

(

f

∘

g

)

=

id

A

(1)

(

f

∘

g

)

∘

(

g

-

1

∘

f

-

1

)

=

id

C

(2)

Note that

id

X

denotes the

identity function

on the set

X

.

The two equations given above follow easily from the fact that function composition is associative.

(

g

-

1

∘

f

-

1

)

∘

(

f

∘

g

)

=

g

-

1

∘

(

(

f

-

1

∘

f

)

∘

g

)

=

g

-

1

∘

(

id

B

∘

g

)

=

g

-

1

∘

g

=

id

A

(

f

∘

g

)

∘

(

g

-

1

∘

f

-

1

)

=

f

∘

(

(

g

∘

g

-

1

)

∘

f

-

1

)

=

f

∘

(

id

B

∘

f

-

1

)

=

f

∘

f

-

1

=

id

C

∎

The socks and shoes rule has a natural

generalization

:

Corollary

.

Let

n

be a

positive integer

and

f

1

,

…

,

f

n

be invertible functions such that their composition

f

1

∘

…

∘

f

n

is well defined. Then

f

1

∘

…

∘

f

n

is invertible and

(

f

1

∘

…

∘

f

n

)

-

1

=

f

n

-

1

∘

…

∘

f

1

-

1

.

A sketch of a proof is as follows: Using

induction

on

n

, the socks and shoes rule can be applied with

f

=

f

1

∘

…

∘

f

n

-

1

and

g

=

f

n

.

Title

inverse of composition of functions

Canonical name

InverseOfCompositionOfFunctions

Date of creation

2013-03-22 17:47:47

Last modified on

2013-03-22 17:47:47

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 03-00

Classification

msc 03E20

Classification

msc 97D40

Synonym

socks and shoes rule

Related topic

Function

Related topic

InverseFormingInProportionToGroupOperation

Generated on Thu Feb 8 19:19:15 2018 by

LaTeXML
line segment

line segment

Definition

Suppose

V

is a vector space over

ℝ

or

ℂ

, and

L

is a subset of

V

.
Then

L

is a

line segment

if

L

can be parametrized
as

L

=

{

a

+

t

⁢

b

∣

t

∈

[

0

,

1

]

}

for some

a

,

b

in

V

with

b

≠

0

.

Sometimes one needs to distinguish between open and closed (

http://planetmath.org/Closed

)
line segments. Then one defines a

closed line segment

as above,
and an

open line segment

as a subset

L

that can be
parametrized as

L

=

{

a

+

t

⁢

b

∣

t

∈

(

0

,

1

)

}

for some

a

,

b

in

V

with

b

≠

0

.

If

x

and

y

are two vectors in

V

and

x

≠

y

, then we denote by

[

x

,

y

]

the set connecting

x

and

y

. This is ,

{

α

⁢

x

+

(

1

-

α

)

⁢

y

|

0

≤

α

≤

1

}

. One can easily check that

[

x

,

y

]

is a closed line segment.

Remarks

•

An alternative,

equivalent

, definition is as follows:
A (closed) line segment is a convex hull of two distinct points.

•

A line segment is

connected

, non-empty set.

•

If

V

is a

topological vector space

, then a closed line segment
is a

closed set

in

V

. However, an open line segment is
an open set in

V

if and only if

V

is one-dimensional.

•

More generally than above, the

concept

of a line segment can be
defined in an

ordered geometry

.

Title

line segment

Canonical name

LineSegment

Date of creation

2013-03-22 14:19:01

Last modified on

2013-03-22 14:19:01

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

12

Author

matte (1858)

Entry type

Definition

Classification

msc 03-00

Classification

msc 51-00

Related topic

Interval

Related topic

LinearManifold

Related topic

LineInThePlane

Related topic

CircularSegment

Defines

open line segment

Defines

closed line segment

Generated on Thu Feb 8 19:19:17 2018 by

LaTeXML
linearly ordered

linearly ordered

An

ordering

≤

(or

<

) of

A

is called

linear

or

total

if any two elements of

A

are

comparable

. The pair

(

A

,

≤

)

is then called a

linearly ordered set

.

Title

linearly ordered

Canonical

name

LinearlyOrdered

Date of creation

2013-03-22 12:09:15

Last modified on

2013-03-22 12:09:15

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

6

Author

akrowne (2)

Entry type

Definition

Classification

msc 03-00

Synonym

linear ordering

Synonym

totally ordered

Synonym

total ordering

Related topic

chain

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

LaTeXML
onto linear transformation, definition of

onto linear transformation, definition of

The

linear transformation

T

:

ℝ

n

→

ℝ

m

:

T

⁢

(

𝐱

)

=

A

⁢

𝐱

is onto iff

Col

⁡

A

spans

ℝ

m

. However, if the corresponding

homogeneous

system has only the

trivial

solution, the

transformation

is

one-to-one

.

Title

onto linear transformation,

definition

of

Canonical

name

OntoLinearTransformationDefinitionOf

Date of creation

2013-03-22 15:25:30

Last modified on

2013-03-22 15:25:30

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 03-00

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

LaTeXML
operator

operator

Synonym of

mapping

and

function

. Often used to refer to mappings where the domain
and codomain are, in some sense a

space of functions

.

Examples:

differential operator

, convolution

operator

.

In the study of

algebraic systems

such as groups and rings, an operator often refers to a mapping from some cartesian power

A

λ

of a set

A

to the set

A

, where

λ

is a

cardinal

. For example,

multiplication

of

integers

can be thought of as an operator from

ℤ

2

to

ℤ

.

Title

operator

Canonical

name

Operator

Date of creation

2013-03-22 12:20:26

Last modified on

2013-03-22 12:20:26

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

6

Author

rmilson (146)

Entry type

Definition

Classification

msc 03-00

Synonym

mapping function

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

LaTeXML
ordered pair

ordered pair

For any sets

a

and

b

, the

ordered pair

(

a

,

b

)

is the set

{

{

a

}

,

{

a

,

b

}

}

.

The characterizing

property

of an ordered pair is:

(

a

,

b

)

=

(

c

,

d

)

⇔

a

=

b

⁢

and

⁢

c

=

d

,

and the above construction of ordered pair, as weird as it seems, is actually the simplest possible formulation which achieves this property.

Title

ordered pair

Canonical

name

OrderedPair

Date of creation

2013-03-22 11:48:53

Last modified on

2013-03-22 11:48:53

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 03-00

Classification

msc 70A05

Classification

msc 70G99

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

LaTeXML
ordered tuplet

ordered tuplet

The

concept

of ordered

n

-tuplet is the

generalization

of

ordered pair

to

n

items. For small values of

n

, the following are used:

n

𝑛𝑎𝑚𝑒

𝑒𝑥𝑎𝑚𝑝𝑙𝑒

3

triplet

(

a

,

b

,

c

)

4

quadruplet

(

a

,

b

,

c

,

d

)

5

quintuplet

(

a

,

b

,

c

,

d

,

e

)

6

sextuplet

(

a

,

b

,

c

,

d

,

e

,

f

)

7

septuplet

(

a

,

b

,

c

,

d

,

e

,

f

,

g

)

8

octuplet

(

a

,

b

,

c

,

d

,

e

,

f

,

g

,

h

)

9

nonuplet

(

a

,

b

,

c

,

d

,

e

,

f

,

g

,

h

,

i

)

10

decuplet

(

a

,

b

,

c

,

d

,

e

,

f

,

g

,

h

,

i

,

j

)

This notion can be defined set-theoretically in a number of ways. For convenience, we shall express two of these definitions for

quintuplets

— it is perfectly easy to generalize them to any other value of

n

.

One possibility is to build

n

-tuplets out of nested ordered pairs. In the case of our example

(

a

,

b

,

c

,

d

,

e

)

, the as a nested ordered pair looks like

(

a

,

(

b

,

(

c

,

(

d

,

e

)

)

)

)

.

This form of is used in the programming language LISP.

Another possibility is to define

n

-tuplets as maps. In this way of thinking, a quintuplet is a

function

whose domain is the set

{

1

,

2

,

3

,

4

,

5

}

. In the case of our example, the function

f

in question is defined as

f

⁢

(

1

)

=

a

f

⁢

(

2

)

=

b

f

⁢

(

3

)

=

c

f

⁢

(

4

)

=

d

f

⁢

(

5

)

=

e

Especially with the second

interpretation

, one sees that a synonym for ”ordered tuplet” is ”

finite sequence

” or ”list”. For

instance

, a quintuplet can also be regarded as a sequence of five items or a list of five items.

Title

ordered tuplet

Canonical name

OrderedTuplet

Date of creation

2013-03-22 14:55:44

Last modified on

2013-03-22 14:55:44

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

16

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03-00

Synonym

tuplet

Synonym

n

-tuplet

Synonym

n

-tuplets

Synonym

ordered

n

-tuplet

Synonym

-tuplet

Synonym

-tuplets

Synonym

tuple

Synonym

n

-tuple

Synonym

ordered

n

-tupule

Synonym

-tuple

Synonym

finite sequence

Related topic

OrderedPair

Related topic

GeneralizedCartesianProduct

Defines

triplet

Defines

quadruplet

Defines

quintuplet

Defines

sextuplet

Defines

septuplet

Defines

octuplet

Defines

nonuplet

Defines

decuplet

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

LaTeXML
ordering relation

ordering relation

Let

S

be a set. An

ordering relation

is a

relation

≤

on

S

such that, for every

a

,

b

,

c

∈

S

:

•

Either

a

≤

b

, or

b

≤

a

,

•

If

a

≤

b

and

b

≤

c

, then

a

≤

c

,

•

If

a

≤

b

and

b

≤

a

, then

a

=

b

.

Equivalently, an ordering relation is a relation

≤

on

S

which makes the pair

(

S

,

≤

)

into a

totally ordered set

.

Warning:

In some cases, an author may use the term “ordering relation” to mean a

partial order

instead of a

total order

.

Given an ordering relation

≤

, one can define a relation

<

by:

a

<

b

if

a

≤

b

and

a

≠

b

. The

opposite

ordering

is the relation

≥

given by:

a

≥

b

if

b

≤

a

, and the relation

>

is defined analogously.

Title

ordering relation

Canonical name

OrderingRelation

Date of creation

2013-03-22 11:52:04

Last modified on

2013-03-22 11:52:04

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 03-00

Classification

msc 81-00

Classification

msc 18-00

Classification

msc 17B37

Classification

msc 18D10

Classification

msc 18D35

Classification

msc 16W30

Related topic

TotalOrder

Related topic

PartialOrder

Related topic

Relation

Defines

opposite ordering

Generated on Thu Feb 8 19:19:29 2018 by

LaTeXML
partition

partition

A

partition

P

of a set

S

is a

collection

of

pairwise disjoint

nonempty sets

such that

∪

P

=

S

.

Any partition

P

of a set

S

introduces an

equivalence relation

on

S

, where each

A

∈

P

is an

equivalence class

. Similarly, given an equivalence relation on

S

, the collection of distinct equivalence classes is a partition of

S

.

Title

partition

Canonical

name

Partition

Date of creation

2013-03-22 11:49:05

Last modified on

2013-03-22 11:49:05

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 03-00

Classification

msc 45D05

Synonym

set partition

Related topic

EquivalenceRelation

Related topic

EquivalenceClass

Related topic

BeattysTheorem

Related topic

Coloring

Generated on Thu Feb 8 19:19:31 2018 by

LaTeXML
partition is equivalent to an equivalence relation

partition is equivalent to an equivalence relation

Proposition 1

.

There is a

one-to-one correspondence

between the set

Part

⁡

(

S

)

of

partitions

of

S

and the set

Equiv

⁡

(

S

)

of

equivalence relations

on

S

.

Proof.

Let

S

be a set.

Suppose

P

=

{

P

i

∣

i

∈

I

}

is a partition of

S

. Form

E

=

⋃

{

P

i

×

P

i

∣

i

∈

I

}

. Given any

a

∈

S

,

a

∈

P

i

for some

i

∈

I

. Then

(

a

,

a

)

∈

P

i

×

P

i

∈

E

. If

(

a

,

b

)

∈

E

, then

(

a

,

b

)

∈

P

i

×

P

i

for some

i

∈

I

, so

a

,

b

∈

P

i

, whence

(

b

,

a

)

∈

P

i

×

P

i

⊆

E

. Finally, suppose

(

a

,

b

)

,

(

b

,

c

)

∈

E

. Then

(

a

,

b

)

∈

P

i

×

P

i

and

(

b

,

c

)

∈

P

j

×

P

j

, or

a

,

b

∈

P

i

and

b

,

c

∈

P

j

. Since

b

∈

P

i

∩

P

j

,

P

i

=

P

j

since

P

is a partition. So

a

,

c

∈

P

i

=

P

j

, or

(

a

,

c

)

∈

P

i

×

P

i

⊆

E

.

Conversely

, suppose

E

is an equivalence relation on

S

. For each

a

∈

S

, define

[

a

]

=

{

b

∈

S

∣

(

a

,

b

)

∈

E

}

.

Then each

a

∈

[

a

]

since

E

is

reflexive

. If

b

∈

[

a

]

, then

(

a

,

b

)

∈

E

or

(

b

,

a

)

∈

E

as

E

is

symmetric

. So

a

∈

[

b

]

as well. Next, pick any

c

∈

[

b

]

. Then

(

b

,

c

)

∈

E

. But

(

a

,

b

)

∈

E

. So

(

a

,

c

)

∈

E

since

E

is

transitive

. Therefore

c

∈

[

a

]

, which implies

[

b

]

⊆

[

a

]

. Collect all the information we have so far, this implies

[

a

]

=

[

b

]

. Therefore

P

=

{

[

a

]

∣

a

∈

S

}

forms a partitition of

S

(

P

is being interpreted as a set, not a

multiset

). In fact,

E

=

⋃

{

[

a

]

×

[

a

]

∣

a

∈

S

}

.
∎

From the above proof, we also have

Corollary 1

.

For every partition

P

=

{

P

i

∣

i

∈

I

}

of

S

,

E

=

⋃

{

P

i

2

∣

i

∈

I

}

is an equivalence relation. Conversely, every equivalence relation on

S

can be formed this way.

Title

partition is

equivalent

to an equivalence relation

Canonical name

PartitionIsEquivalentToAnEquivalenceRelation

Date of creation

2013-03-22 16:44:55

Last modified on

2013-03-22 16:44:55

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 03-00

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

LaTeXML
permutation

permutation

A

permutation

of a

finite set

{

a

1

,

a

2

,

…

,

a

n

}

is an

arrangement

of its

elements

.
For example, if

S

=

{

A

,

B

,

C

}

then

A

⁢

B

⁢

C

,

C

⁢

A

⁢

B

,

C

⁢

B

⁢

A

are three different permutations of

S

.

The number of permutations of a set with

n

elements is

n

!

(see the

rule of product

).

A permutation can also be seen as a

bijective function

of a set into itself.
For example, the permutation

A

⁢

B

⁢

C

↦

C

⁢

A

⁢

B

could be seen a

function

f

:

{

A

,

B

,

C

}

→

{

A

,

B

,

C

}

that assigns:

f

⁢

(

A

)

=

C

,

f

⁢

(

B

)

=

A

,

f

⁢

(

C

)

=

B

.

In fact, every bijection of a set into itself gives a permutation, and any permutation gives rise to a bijective function.

Therefore, we can say that there are

n

!

bijective functions from a set with

n

elements into itself.

Using the function approach, it can be proved that any permutation can be expressed as a

composition

of

disjoint

cycles and also as composition of (not necessarily disjoint)

transpositions

.

Moreover, if

σ

=

τ

1

⁢

τ

2

⁢

⋯

⁢

τ

m

=

ρ

1

⁢

ρ

2

⁢

⋯

⁢

ρ

n

are two

factorization

of a permutation

σ

into transpositions, then

m

and

n

must be both even or both odd. So we can label permutations as

even

or

odd

depending on the number of transpositions for any decomposition.

Permutations (as functions) form in general a

non-abelian group

with function composition as

binary operation

called

symmetric group

of order

n

. The subset of

even permutations

becomes a

subgroup

called the

alternating group

of order

n

.

Title

permutation

Canonical name

Permutation

Date of creation

2013-03-22 11:51:45

Last modified on

2013-03-22 11:51:45

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

13

Author

alozano (2414)

Entry type

Definition

Classification

msc 03-00

Classification

msc 20B99

Classification

msc 46L05

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Classification

msc 22A22

Classification

msc 05A05

Related topic

Bijection

Related topic

Function

Related topic

Cycle2

Related topic

CycleNotation

Related topic

OneLineNotationForPermutations

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

LaTeXML
pointwise

pointwise

When

concepts

(properties,

operations

, etc.) on a set

Y

are extended to

functions

f

:

X

⟶

Y

by treating each

function value

f

⁢

(

x

)

in isolation, the
extended concept is often qualified with the word

pointwise

. One example is pointwise convergence
of functions—a

sequence

{

f

n

}

n

=

1

∞

of
functions

X

⟶

Y

converges

pointwise to
a function

f

if

lim

n

→

∞

⁡

f

n

⁢

(

x

)

=

f

⁢

(

x

)

for all

x

∈

X

.

An important of pointwise concepts
are the

pointwise operations

—operations defined
on functions by applying the operations to function values
separately for each point in the

domain of definition

. These
include

(

f

+

g

)

⁢

(

x

)

=

f

⁢

(

x

)

+

g

⁢

(

x

)

(pointwise addition)

(

f

⋅

g

)

⁢

(

x

)

=

f

⁢

(

x

)

⋅

g

⁢

(

x

)

(pointwise

multiplication

)

(

λ

⁢

f

)

⁢

(

x

)

=

λ

⋅

f

⁢

(

x

)

(pointwise multiplication by scalar)

where the

identities

hold for all

x

∈

X

. Pointwise
operations inherit such properties as

associativity

,

commutativity

,
and distributivity from corresponding operations on

Y

.

An example of an operation on functions which is

not

pointwise is the convolution (

http://planetmath.org/Convolution

)

product

.

Title

pointwise

Canonical name

Pointwise

Date of creation

2013-03-22 15:25:00

Last modified on

2013-03-22 15:25:00

Owner

lars_h (9802)

Last modified by

lars_h (9802)

Numerical id

4

Author

lars_h (9802)

Entry type

Definition

Classification

msc 03-00

Classification

msc 08-00

Defines

pointwise operation

Defines

pointwise addition

Defines

pointwise muliplication

Generated on Thu Feb 8 19:19:46 2018 by

LaTeXML