operationOperation2005-01-19 16:29:522006-10-26 12:28:05Definitionclosed under% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{dfn}{Definition}According to the dictionary \textbf{Webster's 1913}, which can be accessed through
\htmladdnormallink{HyperDictionary.com}{http://www.hyperdictionary.com/}, mathematical
meaning of the word \textit{operation} is: ``\textit{some \textbf{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:
%
\begin{dfn}
\textbf{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.
$$
\# \colon X_1\times X_2\times\cdots\times X_n \longrightarrow X.
$$
\end{dfn}
%
\noindent
Result of operation is usually denoted by one of the following notation:
\begin{itemize}
\item $x_1 \# x_2 \# \cdots \# x_n$
\item $\#(x_1,\ldots,x_n)$
\item $(x_1,\ldots,x_n)_\#$
\end{itemize}
%
The following examples show variety of the concept operation used in mathematics.
\vspace{0.5cm}
\noindent
\underline{\textbf{Examples}}
\begin{enumerate}
\item \textit{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.
\item Operations on vectors in the plane ($\mathbb{R}^2$).
\begin{itemize}
\item \textit{Multiplication by a scalar}. Generalization leads to vector spaces.
\item \textit{Scalar product}. Generalization leads to Hilbert spaces.
\end{itemize}
\item Operations on vectors in the space ($\mathbb{R}^3$).
\begin{itemize}
\item \textit{Cross product}. Can be generalized for the vector space of arbitrary
finite dimension, see vector product in general vector spaces.
\item \textit{Triple product}.
\end{itemize}
\item Some operations on functions.
\begin{itemize}
\item \textit{Composition}.
\item \textit{Function inverse}.
\end{itemize}
\end{enumerate}
\vspace{0.5cm}
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 \textit{closed under} this operation. Formally it is expressed in the following definition.
\begin{dfn}
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 $1\leq j_1 < j_2 < \cdots < j_k\leq n$
such that $X_{j_1}=X_{j_2}=\cdots=X_{j_k}=X$. For simplicity, let us assume that $j_i=i$.
A subset $U\subset X$ is said to be \textbf{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:
$$
\#(u_1,u_2,\ldots,u_k,x_{k+1},x_{k+2},\ldots,x_n)\in U.
$$
\end{dfn}
%
\noindent
The next examples illustrates this definition.
\vspace{0.5cm}
\noindent
\underline{\textbf{Examples}}
\begin{enumerate}
\item Vector space $V$ over a field $K$ is a set, on which the following two operations
are defined:
\begin{itemize}
\item multiplication by a scalar:
$$
\cdot\colon K\times V\longrightarrow V
$$
\item addition
$$
+\colon V\times V \longrightarrow V.
$$
\end{itemize}
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.
\item 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.
\end{enumerate}