operation

operation

According to the dictionary

Webster’s 1913

, which can be accessed through

\htmladdnormallink

HyperDictionary.comhttp://www.hyperdictionary.com/, mathematical
meaning of the word

operation

is: “

some

transformation

to be
made upon quantities

”. Thus, operation is

similar

to

mapping

or

function

. The most
general

mathematical definition

of operation can be made as follows:

Definition 1

Operation

#

defined on the sets

X

1

,

X

2

,

…

,

X

n

with values in

X

is a mapping from

Cartesian product

X

1

×

X

2

×

⋯

×

X

n

to

X

, i.e.

#

:

X

1

×

X

2

×

⋯

×

X

n

⟶

X

.

Result of operation is usually denoted by one of the following notation:

•

x

1

⁢

#

⁢

x

2

⁢

#

⁢

⋯

⁢

#

⁢

x

n

•

#

⁢

(

x

1

,

…

,

x

n

)

•

(

x

1

,

…

,

x

n

)

#

The following examples show

variety

of the

concept

operation used in mathematics.

Examples

1.

Arithmetic operations

:

addition

(

http://planetmath.org/Addition

),

subtraction

,

multiplication

(

http://planetmath.org/Multiplication

),

division

.
Their

generalization

leads to the so-called

binary operations

, which is a basic concept
for such

algebraic structures

as groups and rings.

2.

Operations on vectors in the plane (

ℝ

2

).

–

Multiplication by a scalar

. Generalization leads to

vector spaces

.

–

Scalar product

. Generalization leads to Hilbert spaces.

3.

Operations on vectors in the space (

ℝ

3

).

–

Cross product

. Can be generalized for the vector space of arbitrary
finite

dimension

, see

vector product in general vector spaces

.

–

Triple product

.

4.

Some operations on functions.

–

Composition

.

–

Function

inverse

.

In the case when some of the sets

X

i

are equal to the values set

X

, it is usually said
that operation is defined just on

X

. For such operations, it could be interesting
to consider their action on some subset

U

⊂

X

. In particular,
if operation on

elements

from

U

always gives an element from

U

, it is said that

U

is

closed under

this operation. Formally it is expressed in the following definition.

Definition 2

Let operation

#

:

X

1

×

X

2

×

⋯

×

X

n

⟶

X

is defined
on

X

, i.e. there exists

k

≥

1

and indexes

1

≤

j

1

<

j

2

<

⋯

<

j

k

≤

n

such that

X

j

1

=

X

j

2

=

⋯

=

X

j

k

=

X

. For simplicity, let us assume that

j

i

=

i

.
A subset

U

⊂

X

is said to be

closed under

operation

#

if
for all

u

1

,

u

2

,

…

,

u

k

from U and for all

x

j

∈

X

j

⁢

j

>

k

holds:

#

⁢

(

u

1

,

u

2

,

…

,

u

k

,

x

k

+

1

,

x

k

+

2

,

…

,

x

n

)

∈

U

.

The next examples illustrates this definition.

Examples

1.

Vector space

V

over a field

K

is a set, on which the following two operations
are defined:

–

multiplication by a scalar:

⋅

:

K

×

V

⟶

V

–

addition

+

:

V

×

V

⟶

V

.

Of course these operations need to

satisfy

some

properties

(for details see the entry vector space).
A subset

W

⊂

V

, which is closed under these operations, is called

vector subspace

.

2.

Consider

collection

of all subsets of the real numbers

ℝ

, which we denote by

2

ℝ

.
On this collection, binary operation

intersection of sets

is defined:

∩

:

2

ℝ

×

2

ℝ

⟶

2

ℝ

.

Collection of sets

ℭ

⊂

2

ℝ

:

ℭ

:=

{

[

a

,

b

)

:

a

≤

b

}

is closed under this operation.

Title

operation

Canonical

name

Operation

Date of creation

2013-03-22 14:57:23

Last modified on

2013-03-22 14:57:23

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03E20

Related topic

Function

Related topic

Mapping

Related topic

Transformation

Related topic

BinaryOperation

Defines

closed under

Defines

arithmetic operation

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