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

'''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 relation]]s.

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'' $A$.
<br>
 
$A$ 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.
<br>
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 $a$.

If two mappings $ f_1 = (A_1,B_1,G_1) $ and $ f_2 = (A_2,B_2,G_2) $ satisfy
: $ A_1 \subset A_2 $, $ B_1 \subset B_2 $ and $ G_1 \subset G_2 $
then $f_2$ is called an ''extension'' of $ f_1 $, and $ f_1 $ a ''restriction'' of $f_2$.
In this case, $ f_1 $ is often denoted as $ f_2 \vert A_1 $ 
and, clearly, $ f_1 (a) = f_2 (a) $ holds for all $ a \in A_1 $.

''Remark:''<br>
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.
<br>
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 [[surjection|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,D,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:'' <br>
(a) The condition $ B \subset C $ can be relaxed to $ f(A) \subset C $.
<br>
(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.

====References====
N. Bourbaki,   "Elements of mathematics. Theory of sets" , Addison-Wesley  (1968)  (Translated from French)

Paul R. Halmos, ''Naive Set Theory.''
<br>&nbsp;&nbsp; (The University Series in Undergraduate Mathematics) Princeton, N. J., etc., Van Nostrand, 1960.
<br>&nbsp;&nbsp; ''Reprinted'': (Undergraduate Texts in Mathematics) New York, etc., Springer, 1974.