§l. Set Theory
21
of JR (the set of real numbers) into the set JR is not the same as the map
x t--+ X2 of JR into JR; their graphs are different. Moreover, they do not have
the same properties: in the first case the equation f(x) = b, for a given b E JR
has one or no solution, while it can have two in the second case. Neglecting
these "details" leads to confusion and errors of reasoning.
If A is a subset of a set X, the characteristic function of A (relative to
X) is the map XA : X --+ {O, I} given by
XA(X) = 1 if x E A, = ° if x ¢ A.
If X = JR, one can sketch its graph easily if A is, for example, the union of
a finite number of pairwise disjoint intervals - it consists of horizontal line
segments, with "jumps" at the extremities of these intervals - but you will
not manage if A = Q, the case Dirichlet had spoken of already in about 1830.
The principal interest of these functions is to transform relations between sets
into relations between functions, for example:
etc.
XAnB(X)
XAUB(X)
XX-A(X)
XA(X)xB(X),
XA(X) + XB(X) - XA(X)XB(X),
1 - XA(X),
Instead of speaking of functions one often speaks in mathematics of families of numbers, sets, etc. The only difference between these concepts relates
to the notation employed: given two sets I and X a family of elements of X
indexed by I, the notation is
(Xi)iEI ,
consists of associating an Xi E X to each index i E I; the preceding notation
is thus just another way of speaking of the map i t-------> Xi of I into X, i.e. of
the map f : I --+ X given by
f(i) = Xi for every i E I.
One might do entirely without this concept, whose historical origin lies in
sequences of real numbers
which we will meet from the beginning of the next chapter, for example the
sequence
1,1/2, . .. , lin, . .. ;
for clasSIcal analysts, who concerned themselves only with functions where
the variable can take all the real values in an interval, the notation Un denotes
term number n in the sequence; but you can, without the least inconvenience,
21
of JR (the set of real numbers) into the set JR is not the same as the map
x t--+ X2 of JR into JR; their graphs are different. Moreover, they do not have
the same properties: in the first case the equation f(x) = b, for a given b E JR
has one or no solution, while it can have two in the second case. Neglecting
these "details" leads to confusion and errors of reasoning.
If A is a subset of a set X, the characteristic function of A (relative to
X) is the map XA : X --+ {O, I} given by
XA(X) = 1 if x E A, = ° if x ¢ A.
If X = JR, one can sketch its graph easily if A is, for example, the union of
a finite number of pairwise disjoint intervals - it consists of horizontal line
segments, with "jumps" at the extremities of these intervals - but you will
not manage if A = Q, the case Dirichlet had spoken of already in about 1830.
The principal interest of these functions is to transform relations between sets
into relations between functions, for example:
etc.
XAnB(X)
XAUB(X)
XX-A(X)
XA(X)xB(X),
XA(X) + XB(X) - XA(X)XB(X),
1 - XA(X),
Instead of speaking of functions one often speaks in mathematics of families of numbers, sets, etc. The only difference between these concepts relates
to the notation employed: given two sets I and X a family of elements of X
indexed by I, the notation is
(Xi)iEI ,
consists of associating an Xi E X to each index i E I; the preceding notation
is thus just another way of speaking of the map i t-------> Xi of I into X, i.e. of
the map f : I --+ X given by
f(i) = Xi for every i E I.
One might do entirely without this concept, whose historical origin lies in
sequences of real numbers
which we will meet from the beginning of the next chapter, for example the
sequence
1,1/2, . .. , lin, . .. ;
for clasSIcal analysts, who concerned themselves only with functions where
the variable can take all the real values in an interval, the notation Un denotes
term number n in the sequence; but you can, without the least inconvenience,