§l. Set Theory
19
numbers as subsets of P(Q) and proved their fundamental properties one of
course forgets their manifestly complicated definition.
One also uses P in the study of equivalence relations. Such is any relation
R (denoted for example by xRy) between the elements of a set X satisfying
the following conditions: (i) xRy and yRz imply xRz, (ii) xRy implies yRx,
(iii) xRx is true for all x E X; consider for instance the relation "x - y is
a multiple of 3" between signed integers. If a E X the axiom of separation
allows one to speak of the set C(a) of all x E X such that xRa is true; this is
called the equivalence class of a. It is immediate that a E C(a) and that two
classes C(a) and C(b) are either identical or disjoint. The axiom of the set
of subsets allows us to consider the C(a), a E X, as the elements of a new
set, contained in P(X), denoted XI R, and called the quotient of X by the
equivalence relation R. See for example the short §4 of my Algebra, where
you will find examples and also applications to the construction of the signed
integers and of the rational numbers. Observe in passing that, by explicit
construction, and not just from the general abstract theory, a quotient set is
a set of sets.
5 - Functions, maps, correspondences
The concept of the cartesian product allows one to introduce the general
concept of a function or map, which is as fundamental as that of a set and
which, as we shall see, reduces to it as do all others. In elementary education
and in the whole history of mathematics up to the beginning of the XIXth
century, a function was given by a "formula" such as f(x) = x 2 - 3, f(x) =
sin x, etc., but starting with Descartes one often also defined a function from a
curve whose "equation" one sought. For experimental scientists and engineers
a function is very often also given by its graph, the geometrical locus of those
points (x, y) in the plane such that y = f(x) for a function f which, quite
often, one does not really know.
Starting with the XIX th century the concept of a function ceased to be
associated with a simple or complicated "formula"; the German Dirichlet for
example speaks of the function equal to 0 if x is a rational number and to 1
if x is irrational, and one later envisaged much stranger functions, until the
general and abstract concept emerged of a function defined on a set X and
having values in a set Y; such a function f associates to every x E X a well
determined y = f(x) E Y depending on x according to a precise rule. The
graph of f is then the set of ordered pairs (x, y) E X x Y such that y = f(x)
for every x EX. One encounters this in everyday life: if, in a monogamous
society, one denotes by H the set of married men and by F the set of women,
the relation "y is the wife of x" is a function with values in F defined on H.
Its graph is clearly a set of . .. couples.
Conversely, a subset G of X x Y is the graph of a function f provided that
G has the following property: for every x E X there exists one, and only one,
y E Y such that (x, y) E G; and then one writes y = f(x). This convention
19
numbers as subsets of P(Q) and proved their fundamental properties one of
course forgets their manifestly complicated definition.
One also uses P in the study of equivalence relations. Such is any relation
R (denoted for example by xRy) between the elements of a set X satisfying
the following conditions: (i) xRy and yRz imply xRz, (ii) xRy implies yRx,
(iii) xRx is true for all x E X; consider for instance the relation "x - y is
a multiple of 3" between signed integers. If a E X the axiom of separation
allows one to speak of the set C(a) of all x E X such that xRa is true; this is
called the equivalence class of a. It is immediate that a E C(a) and that two
classes C(a) and C(b) are either identical or disjoint. The axiom of the set
of subsets allows us to consider the C(a), a E X, as the elements of a new
set, contained in P(X), denoted XI R, and called the quotient of X by the
equivalence relation R. See for example the short §4 of my Algebra, where
you will find examples and also applications to the construction of the signed
integers and of the rational numbers. Observe in passing that, by explicit
construction, and not just from the general abstract theory, a quotient set is
a set of sets.
5 - Functions, maps, correspondences
The concept of the cartesian product allows one to introduce the general
concept of a function or map, which is as fundamental as that of a set and
which, as we shall see, reduces to it as do all others. In elementary education
and in the whole history of mathematics up to the beginning of the XIXth
century, a function was given by a "formula" such as f(x) = x 2 - 3, f(x) =
sin x, etc., but starting with Descartes one often also defined a function from a
curve whose "equation" one sought. For experimental scientists and engineers
a function is very often also given by its graph, the geometrical locus of those
points (x, y) in the plane such that y = f(x) for a function f which, quite
often, one does not really know.
Starting with the XIX th century the concept of a function ceased to be
associated with a simple or complicated "formula"; the German Dirichlet for
example speaks of the function equal to 0 if x is a rational number and to 1
if x is irrational, and one later envisaged much stranger functions, until the
general and abstract concept emerged of a function defined on a set X and
having values in a set Y; such a function f associates to every x E X a well
determined y = f(x) E Y depending on x according to a precise rule. The
graph of f is then the set of ordered pairs (x, y) E X x Y such that y = f(x)
for every x EX. One encounters this in everyday life: if, in a monogamous
society, one denotes by H the set of married men and by F the set of women,
the relation "y is the wife of x" is a function with values in F defined on H.
Its graph is clearly a set of . .. couples.
Conversely, a subset G of X x Y is the graph of a function f provided that
G has the following property: for every x E X there exists one, and only one,
y E Y such that (x, y) E G; and then one writes y = f(x). This convention