Function composition
The term composition is usually in the mathematics of the series connection function, also known as chaining, or one behind the other embodiments referred to. It is usually recorded using the concatenation character.
The representation of a function as a concatenation of two or more, generally simpler functions is important, for example in differential and integral calculus, when it comes to compute derivatives with the chain rule or integral with the substitution rule.
The term composition can be generalized from functions to relations and partial functions.
- 5.1 Examples
- 8.1 Example
- 8.2 properties
Definition
Be arbitrary sets and functions, and so the function is called
The composition of and or after. We then say well composed with. It should be noted here that the mapping is applied first to the right, as opposed to the diagram
Different spellings
An alternative way of writing, where you this should not be confused with the product of functions in which the multiplication sign is also often omitted.
There are also a few authors who write after than, that evaluate the functions from left to right. What order is selected, can often understand with an example of the author. In addition, the notation, in which the function symbol is written to the right of the argument, ie (or ) instead of exists. Then the evaluation is from left to right is obvious, so (mainly in the context of ( right ) group operations spread ).
Examples
Consider the following functions for which a definition and set of values , the set of real numbers or a subset of it is assumed. Is the function and the function given by, the result of the concatenation and the function with
Conversely, the function defined by the group, at
Are.
Properties
Associativity
The composition of functions is associative, ie for functions, and the following applies:
Because
Commutativity
The composition of functions is not commutative in general; example applies to the functions and:
Identical pictures
The identity map behaves in the composition of neutral, for a function so the following applies:
Where and are the respective identities on the quantities and present.
Injectivity, surjectivity, bijectivity
Important properties which may have a function are
- Injectivity (no element is assumed multiple),
- Surjective ( every element in is accepted)
- Bijectivity ( each element in is accepted, and none is believed more than once).
Each of these properties is transferred to the concatenation, it is thus:
- The composition of injective functions is injective.
- The composition of surjective functions is surjective.
- The composition of bijective functions is bijective.
Conversely, is a concatenation
- Injective, then is injective.
- Surjective, then is surjective.
- Bijective, so is injective and surjective.
Potencies ( iteration)
Is a mapping of a set into itself, then you can link this function with itself and receives. As usual with associative operations, the -th power can now inductively for every natural number can be explained by:
In addition, one sets
Is also referred to as a -th iterate of; the (even multiple ) concatenating a function with itself is called iteration.
If it is defined on a multiplication, the iteration may not be confused with the multiplication can in this case describe the expression.
Is even bijective, then there exists the inverse function, and the negative powers are defined by:
Examples
Let the set of positive real numbers and given by. Then:
Algebraic Structures
If the set of all functions from a given quantity considered in themselves, the composition defined on an internal binary operation in respect of which ( with the identity map as a neutral element ) is a monoid.
If only bijective functions used, the monoid is even a group with the corresponding inverse function as an inverse element. If the set is finite, it is a symmetric group.
Structure compatible pictures
In mathematics one often considered quantities with an additional structure as well as images that are compatible with this structure, for example
- Linear maps between vector spaces
- Continuous maps between topological spaces
- Group homomorphisms between groups
It is desirable now that the structure compatibility is maintained in the composition, and in fact is true in the examples:
- The composition of linear maps is linear.
- The composition of continuous maps is continuous.
- The composition of group homomorphisms is a group homomorphism.
These considerations lead to the category theory, in which one of them even abstracted that it is pictures, and only the associativity and the property of the identities for the composition calls.
Composition of relations
A -function of the graph is a ratio. Regarding the composition of functions then applies (using infix notation ):
This observation leads to the definition of the composition of two binary relations and: The relation is given by
In the composition of relations that is the order from right to left is always respected.
Example
Be the set of points, the set of lines and the amount of layers in the three-dimensional space. The relations and are defined by:
Then for the composition:
Properties
- The composition of relations is associative.
- Identifies the identical relation on a set, ie the set of all pairs, then for each relation:
- Is a relation on a set, then so also all powers are (also) defined. These powers are used for example in the definition of reflexive - transitive closure. A relation is called with transitive.
Different notation in physics
In physics and other sciences, it is common to identify the concatenation of a function with the " outer function ". Because this notation arise in physical literature partially equations that are false or meaningless at first glance, according to common mathematical conventions about
Where the position vector of the point and its Euclidean length. This equation is mathematically, in principle wrong, because according to the left-hand side of the equation is a function (it is still in an element ), on the right side apparently as a domain of a subset of the real numbers has, therefore, since in the scalar quantity used. What is meant with this intuitive equation, however, that (for a special case under consideration ) the physical size (in this case, a potential ), the iA a function of position, it can be described by a function that depends only on the distance of the place from the zero point. A mathematically "clean" formulation of this statement would be something like:
Is thus a concatenation of the scalar function and the Euclidean norm:
Again identify with the outer function and this has the potential - we get the above, intuitive notation of this equation by first chaining - symbolic. Advantages of the notation are intuitive and writing a small number of different symbols. A typical example of a function, which satisfies the above equation, is the central potential of the form
Which, inter alia, be used in electrostatics. in this case, a chain of the scalar function
With the Euclidean norm: