According to the dictionary Webster’s 1913, which can be accessed through \htmladdnormallinkHyperDictionary.comhttp://www.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:
Operation defined on the sets with values in is a mapping from Cartesian product to , i.e.
Result of operation is usually denoted by one of the following notation:
Some operations on functions.
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 . 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.
Let operation is defined on , i.e. there exists and indexes such that . For simplicity, let us assume that . A subset is said to be closed under operation if for all from U and for all holds:
The next examples illustrates this definition.
Vector space over a field is a set, on which the following two operations are defined:
multiplication by a scalar:
|Date of creation||2013-03-22 14:57:23|
|Last modified on||2013-03-22 14:57:23|
|Last modified by||rspuzio (6075)|