Mapping

$ \def\P{\mathcal P} % power set \def\iff{\Longleftrightarrow} $

Mapping, or abbreviated map, is one of many synonyms used for function. In particular, the term map(ping) is used in general contexts, such as set theory, but usage is not restricted to these cases.

The mapping concept in set theory

In set theory mappings are special binary relations.

A mapping $f$ from a set $A$ to a set $B$ is an (ordered) triple $ f = (A,B,G_f) $ where $ G_f \subset A \times B $ such that

• (a) if $ (x,y) $ and $ (x,y') \in G_f $ then $ y=y' $, and
• (b) the projection $ \pi_1 (G_f) = \{ x \mid (x,y) \in G_f \} = A $.

Condition (a) expresses that $f$ is single-valued. and condition (b) that it is defined on $A$.

$A$ is the domain, $B$ is the codomain, and $G_f$ is the graph of the mapping. Therefore, in this setting, mappings are equal if and only if all three corresponding components (domain, codomain, and graph) are equal.
The mapping is usually denoted as $ f : A \to B $, and $ a \mapsto f(a) $ where $ f(a) := b \iff (a,b) \in G_f $ is the value of $f$ at $a$.

Remark:
Sometimes only the graph $G_f$ is used to represent a function. In this case two mappings are equal if they have the same graph, and one may allow graphs that are not sets but classes.
While the domain of the function can be obtained as projection $ \pi_1 (G_f) $ of the first component, the projection $ \pi_2 (G_f) $ of the second component does not produce the codomain but only the image of the domain. Thus the concept of surjectivity is not applicable.

Composition

Two mappings can be composed if the codomain of one mapping is a subset of the domain of the other mapping:

For $ f=(A,B,G_f) $ and $ g=(C,D,G_g) $ with $ B \subset C $ the composition $ g \circ f $ is the mapping $ (A,C,G) $ with

$ G := \{ (a,g(f(a))) \mid a \in A \} = \{ (a,c) \mid (\exists b \in B) (a,b) \in G_f \land (b,c) \in G_g \} $.

Remarks:
(a) The condition $ B \subset C $ can be relaxed to $ f(B) \subset C $.
(b) If only graphs are used then the graph of the composition is defined (as above) by

$ G_{g \circ f} := \{ (a,c) \mid (\exists b ) (a,b) \in G_f \land (b,c) \in G_g \} $

but may turn out to be empty.

Induced mappings

Every mapping $ f : A \to B $ induces two mappings between the power sets $\P(A)$ and $\P(B)$.

$ f_\ast : \P(A) \to \P(B) $ defined by $ f_\ast (S) := \{ f(a) \mid a \in S \}$ for $ S \subset A $

and

$ f^\ast : \P(B) \to \P(A) $ defined by $ f^\ast (T) := \{ a \mid f(a) \in T \}$ for $ T \subset B $

$ f_\ast (S) $ is called the image of $S$ under $f$, usually denoted as $f(S)$, and $ f^\ast (T) $ is called the inverse image of $T$ under $f$, usually denoted as $f^{-1}(T)$, but one has to be aware that these common notations may be ambiguous in certain situations.

single-valued

A law according to which to every element of a given set has been assigned a completely defined element of another given set ( may coincide with ). Such a relation between the elements and is denoted in the form , or . One also writes and says that the mapping operates from into . The set is called the domain (of definition) of the mapping, while the set is called the range (of values) of the mapping. The mapping is also called a mapping of the set into the set (or onto the set if ). Logically, the concept of a "mapping" coincides with the concept of a function, an operator or a transformation.

A mapping gives rise to a set , which is called the graph of the mapping. On the other hand, a set defines a single-valued mapping having graph if and only if for all one and only one exists such that ; and then .

Two mappings and are said to be equal if their domains of definition coincide and if for each . In this case the ranges of these mappings also coincide. A mapping defined on is constant if there is an such that for every . The mapping defined on a subset of by the equality , , is called the restriction of the mapping to ; this restriction is often denoted by . A mapping defined on a set and satisfying the equality for all is called an extension (or continuation) of the mapping to . If three sets are given, if a mapping with values in is defined on , and a mapping with values in is defined on , then there exists a mapping with domain of definition , taking values in , and defined by the equality . This mapping is called the composite of the mappings and , while and are called component (factor) mappings. The mapping is also called the compound mapping (composite mapping, composed mapping), consisting of the interior mapping and the exterior mapping . The composed mapping is denoted by , where the order of the notation is vital (for functions of a real variable, the term superposition is also used). The concept of a compound mapping can be generalized to any finite number of components of the mapping.

A mapping , defined on and taking values in , gives rise to a new mapping defined on the subsets of and taking subsets of as values. In fact, if , then

The set is called the image of . If , the initial mapping is obtained; thus, is an extension of from the set to the set of all subsets of if a one-element set is identified with the element comprising it. When , a set is called an invariant subset for if , while a point is called a fixed point for if . Invariant sets and fixed points are important in solving functional equations of the form or .

Every mapping gives rise to a mapping defined on the subsets of the set or and taking subsets of the set as values. In fact, for every (or ), the set is denoted by , and is called the complete inverse image (complete pre-image) of . If for each consists of a single element, then is a mapping of elements, is defined on , and takes values in . It is also called the inverse mapping for . The existence of an inverse mapping is equivalent to the solvability of the equation , , for a unique when is given.

If the sets and have certain properties, then interesting classes can be distinguished in the set of all mappings from into . Thus, for partially ordered sets and , the mapping is isotone if implies (cf. Isotone mapping). For complex planes and , the class of holomorphic mappings is naturally selected. For topological spaces and , the class of continuous mappings between these spaces is distinguished naturally; an extended theory of differentiation of mappings (cf. Differentiation of a mapping) has been constructed. For mappings of a scalar argument and, in the most general case, for mappings defined on a measure space, the concept of (weak or strong) measurability can be introduced, and various Lebesgue-type integrals can be constructed (for example, the Bochner integral and the Daniell integral).

A mapping is called a multi-valued mapping if subsets consisting of more than one element are assigned to certain values of . Examples of this type of mappings include multi-sheeted functions of a complex variable, multi-valued mappings of topological spaces, and others.

References

[1]	N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French)
[2]	N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
[3]	J.L. Kelley, "General topology" , Springer (1975)

Comments

For a mapping , the set is also called the source of , while is also called the target of , [a3].

References

[a1]	P.R. Halmos, "Measure theory" , v. Nostrand (1950)
[a2]	P.R. Halmos, "Naive set theory" , v. Nostrand (1961)
[a3]	P.M. Cohn, "Universal algebra" , Reidel (1981)