operation

According to the dictionary Webster’s 1913, which can be accessed throughHyperDictionary.comhttp://www.hyperdictionary.com/, mathematicalmeaning of the word operation is: “some transformation to bemade upon quantities”. Thus, operation is similar to mapping or function. The mostgeneral 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.

$$\mathrm{\#}:X_1\times X_2\times \mathrm{\cdots }\times X_n⟶X.$$

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 so-called binary operations, which is a basic conceptfor 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 arbitraryfinite 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 saidthat operation is defined just on $X$. For such operations, it could be interestingto 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 $\mathrm{\#}\mathrm{:}{X}_{\mathrm{1}}\mathrm{\times }{X}_{\mathrm{2}}\mathrm{\times }\mathrm{\cdots }\mathrm{\times }{X}_n\mathrm{⟶}X$ is definedon $X$, 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 $\mathrm{\#}$ iffor 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 operationsare defined:

   • multiplication by a scalar:

     $$\cdot :K\times V⟶V$$

   • addition

     $$+:V\times V⟶V.$$

   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.

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:

   $$\cap :2^{ℝ}\times 2^{ℝ}⟶2^{ℝ}.$$

   Collection of sets $ℭ\subset 2^{ℝ}$:

   $$ℭ:=\{[a,b):a\le b\}$$

   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