Download PDF

operation∗
rspuzio†
2013-03-21 18:33:59
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 X1 , X2 , . . . , Xn with values in
X is a mapping from Cartesian product X1 × X2 × · · · × Xn to X, i.e.
# : X1 × X2 × · · · × Xn −→ X.
Result of operation is usually denoted by one of the following notation:
• x1 #x2 # · · · #xn
• #(x1 , . . . , xn )
• (x1 , . . . , xn )#
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 (R2 ).
• Multiplication by a scalar. Generalization leads to vector spaces.
• Scalar product. Generalization leads to Hilbert spaces.
3. Operations on vectors in the space (R3 ).
∗ hOperationi
created: h2013-03-21i by: hrspuzioi version: h36653i Privacy setting: h1i
hDefinitioni h03E20i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1
• 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 Xi 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 ⊂ 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 # : X1 × X2 × · · · × Xn −→ X is defined on X,
i.e. there exists k ≥ 1 and indexes 1 ≤ j1 < j2 < · · · < jk ≤ n such that
Xj1 = Xj2 = · · · = Xjk = X. For simplicity, let us assume that ji = i. A
subset U ⊂ X is said to be closed under operation # if for all u1 , u2 , . . . , uk
from U and for all xj ∈ Xj j > k holds:
#(u1 , u2 , . . . , uk , xk+1 , xk+2 , . . . , xn ) ∈ U.
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:
· : K × V −→ V
• addition
+ : V × V −→ V.
Of course these operations need to satisfy some properties (for details see
the entry vector space). A subset W ⊂ V , which is closed under these
operations, is called vector subspace.
2. Consider collection of all subsets of the real numbers R, which we denote
by 2R . On this collection, binary operation intersection of sets is defined:
∩ : 2R × 2R −→ 2R .
Collection of sets C ⊂ 2R :
C := {[a, b) : a ≤ b}
is closed under this operation.
2