operation

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:

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. 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 groups and rings.
2. Operations on vectors in the plane ($\mathbb{R}^2$ ).
   • Multiplication by a scalar. Generalization leads to vector spaces.
   • Scalar product. Generalization leads to Hilbert spaces.
3. Operations on vectors in the space ($\mathbb{R}^3$ ).
   • Cross product. Can be generalized for the vector space of arbitrary finite 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 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.

The next examples illustrates this definition.

Examples

1. Vector space $V$ over a field $K$ is a set, on which the following two operations are defined:
   • multiplication by a scalar: $$ \cdot\colon K\times V\longrightarrow V $$
   • 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.
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.