operation
According to the dictionary Webster’s 1913, which can be accessed through \htmladdnormallinkHyperDictionary.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 $\mathrm{\#}$ defined on the sets ${\mathrm{X}}_{\mathrm{1}}\mathrm{,}{\mathrm{X}}_{\mathrm{2}}\mathrm{,}\mathrm{\dots}\mathrm{,}{\mathrm{X}}_{\mathrm{n}}$ with values in $\mathrm{X}$ is a mapping from Cartesian product ${\mathrm{X}}_{\mathrm{1}}\mathrm{\times}{\mathrm{X}}_{\mathrm{2}}\mathrm{\times}\mathrm{\cdots}\mathrm{\times}{\mathrm{X}}_{\mathrm{n}}$ to $\mathrm{X}$, i.e.
$$\mathrm{\#}:{X}_{1}\times {X}_{2}\times \mathrm{\cdots}\times {X}_{n}\u27f6X.$$ 
Result of operation is usually denoted by one of the following notation:

•
${x}_{1}\mathrm{\#}{x}_{2}\mathrm{\#}\mathrm{\cdots}\mathrm{\#}{x}_{n}$

•
$\mathrm{\#}({x}_{1},\mathrm{\dots},{x}_{n})$

•
${({x}_{1},\mathrm{\dots},{x}_{n})}_{\mathrm{\#}}$
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 socalled binary operations^{}, which is a basic concept for such algebraic structures^{} as groups and rings.

2.
Operations on vectors in the plane (${\mathbb{R}}^{2}$).

–
Multiplication by a scalar. Generalization leads to vector spaces^{}.

–
Scalar product^{}. Generalization leads to Hilbert spaces.

–

3.
Operations on vectors in the space (${\mathbb{R}}^{3}$).

–
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 $U\subset X$. 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 $\mathrm{\#}\mathrm{:}{X}_{\mathrm{1}}\mathrm{\times}{X}_{\mathrm{2}}\mathrm{\times}\mathrm{\cdots}\mathrm{\times}{X}_{n}\mathrm{\u27f6}X$ is defined on $X$, i.e. there exists $k\mathrm{\ge}\mathrm{1}$ and indexes $$ such that ${X}_{{j}_{\mathrm{1}}}\mathrm{=}{X}_{{j}_{\mathrm{2}}}\mathrm{=}\mathrm{\cdots}\mathrm{=}{X}_{{j}_{k}}\mathrm{=}X$. For simplicity, let us assume that ${j}_{i}\mathrm{=}i$. A subset $U\mathrm{\subset}X$ is said to be closed under operation $\mathrm{\#}$ if for all ${u}_{\mathrm{1}}\mathrm{,}{u}_{\mathrm{2}}\mathrm{,}\mathrm{\dots}\mathrm{,}{u}_{k}$ from U and for all ${x}_{j}\mathrm{\in}{X}_{j}\mathit{}j\mathrm{>}k$ holds:
$$\mathrm{\#}({u}_{1},{u}_{2},\mathrm{\dots},{u}_{k},{x}_{k+1},{x}_{k+2},\mathrm{\dots},{x}_{n})\in 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:
$$\cdot :K\times V\u27f6V$$ 
–
addition
$$+:V\times V\u27f6V.$$
Of course these operations need to satisfy some properties (for details see the entry vector space). A subset $W\subset V$, which is closed under these operations, is called vector subspace.

–

2.
Consider collection^{} of all subsets of the real numbers $\mathbb{R}$, which we denote by ${2}^{\mathbb{R}}$. On this collection, binary operation intersection of sets is defined:
$$\cap :{2}^{\mathbb{R}}\times {2}^{\mathbb{R}}\u27f6{2}^{\mathbb{R}}.$$ Collection of sets $\u212d\subset {2}^{\mathbb{R}}$:
$$\u212d:=\{[a,b):a\le b\}$$ is closed under this operation.
Title  operation 
Canonical name  Operation 
Date of creation  20130322 14:57:23 
Last modified on  20130322 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 