PlanetMath: operation

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (39)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

operation

(Definition)

According to the dictionary Webster's 1913, which can be accessed through 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 $\char93$ defined on the sets $X_1,X_2,\ldots,X_n$ with values in is a mapping from Cartesian product $X_1\times X_2\times \cdots\times X_n$ to , i.e.

$\displaystyle \char93 \colon X_1\times X_2\times\cdots\times X_n \longrightarrow X.$

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

$x_1 \char93 x_2 \char93 \cdots \char93 x_n$
$\char93 (x_1,\ldots,x_n)$
$(x_1,\ldots,x_n)_\char93$

The following examples show variety of the concept operation used in mathematics.

Examples

Arithmetic operations: addition, subtraction, multiplication, division. Their generalization leads to the so-called binary operations, which is a basic concept for such algebraic structures as group, ring, field.
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.
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.
Some operations on functions.
- Composition.
- Function inverse.

In the case when some of the sets are equal to the values set , it is usually said that operation is defined just on . For such operations, it could be interesting to consider their action on some subset $U\subset X$ . In particular, if operation on elements from always gives an element from , it is said that is closed under this operation. Formally it is expressed in the following definition.

Definition 2 Let operation $\char93 \colon X_1\times X_2\times\cdots\times X_n \longrightarrow X$ is defined on , i.e. there exists $k\geq 1$ and indexes $1\leq j_1 < j_2 < \cdots < j_k\leq n$ such that $X_{j_1}=X_{j_2}=\cdots=X_{j_k}=X$ . For simplicity, let us assume that . A subset $U\subset X$ is said to be closed under operation $\char93$ if for all $u_1,u_2,\ldots,u_k$ from U and for all $x_j\in X_j\, j>k$ holds:

$\displaystyle \char93 (u_1,u_2,\ldots,u_k,x_{k+1},x_{k+2},\ldots,x_n)\in U.$

The next examples illustrates this definition.

Examples

Vector space over a field is a set, on which the following two operations are defined:
- multiplication by a scalar:
  $\displaystyle \cdot\colon K\times V\longrightarrow V$
- addition
  $\displaystyle +\colon V\times V \longrightarrow V.$
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.
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:
$\displaystyle \cap\colon 2^\mathbb{R} \times 2^\mathbb{R} \longrightarrow 2^\mathbb{R}.$
Collection of sets $\mathfrak{C}\subset 2^\mathbb{R}$ :
$\displaystyle \mathfrak{C}:=\{ [a,b) \colon \, a\leq b \}$
is closed under this operation.