operation

According to the dictionary Webster’s 1913, which can be accessed through \htmladdnormallinkHyperDictionary.comhttp://www.hyperdictionary.com/, mathematical meaning of the word is: “some 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 $X_{1},X_{2},\ldots,X_{n}$ with values in $X$ is a mapping from Cartesian product $X_{1}\times X_{2}\times\cdots\times X_{n}$ to $X$, i.e.

 $\#\colon X_{1}\times X_{2}\times\cdots\times X_{n}\longrightarrow X.$

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

• $x_{1}\#x_{2}\#\cdots\#x_{n}$

• $\#(x_{1},\ldots,x_{n})$

• $(x_{1},\ldots,x_{n})_{\#}$

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

Examples

1. 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. 2.

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

• Multiplication by a scalar. Generalization leads to vector spaces.

• . Generalization leads to Hilbert spaces.

3. 3.

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

4. 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 $\#\colon X_{1}\times X_{2}\times\cdots\times X_{n}\longrightarrow X$ is defined on $X$, i.e. there exists $k\geq 1$ and indexes $1\leq j_{1} such that $X_{j_{1}}=X_{j_{2}}=\cdots=X_{j_{k}}=X$. For simplicity, let us assume that $j_{i}=i$. A subset $U\subset X$ is said to be closed under operation $\#$ if for all $u_{1},u_{2},\ldots,u_{k}$ from U and for all $x_{j}\in X_{j}\,j>k$ holds:

 $\#(u_{1},u_{2},\ldots,u_{k},x_{k+1},x_{k+2},\ldots,x_{n})\in U.$

The next examples illustrates this definition.

Examples

1. 1.

Vector space $V$ over a field $K$ is a set, on which the following two operations are defined:

• multiplication by a scalar:

 $\cdot\colon K\times V\longrightarrow V$

 $+\colon V\times V\longrightarrow V.$

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.

2. 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:

 $\cap\colon 2^{\mathbb{R}}\times 2^{\mathbb{R}}\longrightarrow 2^{\mathbb{R}}.$

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

 $\mathfrak{C}:=\{[a,b)\colon\,a\leq b\}$

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