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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 $\#$ 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
- 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 ($\mathbb{R}^2$ ).
- Operations on vectors in the space ($\mathbb{R}^3$ ).
- Some operations on functions.
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 $1\leq j_1 < j_2 < \cdots < j_k\leq n$ 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 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
- Vector space $V$ over a field $K$ is a set, on which the following two operations are defined:
- multiplication by a scalar: $$ \cdot\colon K\times V\longrightarrow V $$
- addition $$ +\colon V\times V \longrightarrow V. $$
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: $$ \cap\colon 2^\mathbb{R} \times 2^\mathbb{R} \longrightarrow 2^\mathbb{R}. $$ Collection of sets $\mathfrak{C}\subset 2^\mathbb{R}$ : $$ \mathfrak{C}:=\{ [a,b) \colon \, a\leq b \} $$ is closed under this operation.
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"operation" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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Cross-references: intersection of sets, real numbers, collection, vector subspace, properties, satisfy, indexes, elements, subset, action, inverse, vector product in general vector spaces, dimension, finite, Hilbert spaces, product, vector spaces, scalar, plane, vectors, field, ring, group, algebraic structures, binary operations, division, subtraction, variety, Cartesian product, mathematical definition, function, mapping, similar
There are 534 references to this entry.
This is version 5 of operation, born on 2005-01-19, modified 2006-10-26.
Object id is 6653, canonical name is Operation.
Accessed 16710 times total.
Classification:
| AMS MSC: | 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory ) |
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Pending Errata and Addenda
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