20
I - Sets and Functions
allows one to reduce the concept of a function to that of a set: by definition
a function defined on X with values in Y is a subset of X x Y subject to the
preceding condition; no longer is there a "formula".
On suppressing the restriction imposed on G one obtains the concept of a
correspondence or relation between X and Y: two elements x E X and y E Y
correspond under G if (x, y) E G. If one returns to the preceding example and
replaces the set H by the set of all men, and does not assume that the society
is monogamous, the relation "y is one of the wives of x" is a correspondence
between Hand F. One does not insist on the existence for every x E H of
ayE F such that (x, y) E G, nor does one insist that such a y be unique;
the x for which there exists such a y constitute the set of definition of the
correspondence (the owners of a harem); the y such that one has (x, y) E G
for at least one x constitute the image or the set of values (the women of a
harem). If X = Y = JR, the relation x 2 +3y2 = 1, whose graph G in the plane
is an ellipse, is a correspondence: its set of definition is the set of x such that
Ixl S; 1, its image the set of y such that Iyl S; 1/V3; this correspondence is
not a function since a real number may have two distinct square roots. The
formula x < y is likewise a correspondence (one more often says "relation"
in cases of this sort) whose graph the reader will have no trouble finding.
In actual practice one often uses other expressions. Instead of saying
let f be a function defined on X with values in Y,
one often says
let f be a map of X into Y
or
consider a map f : X --+ Y.
When f is given by a "formula" one also, for example, speaks of
the map x ~ x 3 of X into Y,
assuming that this makes sense; do not confuse the signs --+ and ~; the
string x ~ x 3 does not denote a map of the set x into the set. x 3 , it denotes
the function or map which to each element x of X associates the element x 3
ofY.
Let us again observe that in mathematics, when speaking of a function
or map f, one must specify the set X on which f is defined and the set Y
in which it takes its values. To speak without further specification of ''the
function x 2 " is meaningless 21 . The map x ~ x 2 of the interval 0 S; x S; 1
21 The logicians nevertheless speak of functional relations without specifying the
sets of departure or arrival: they mean a relation R{ x, y} between two "variables"
x and y such that
R{x,y'} & R{x,y"} implies y' = y".
I - Sets and Functions
allows one to reduce the concept of a function to that of a set: by definition
a function defined on X with values in Y is a subset of X x Y subject to the
preceding condition; no longer is there a "formula".
On suppressing the restriction imposed on G one obtains the concept of a
correspondence or relation between X and Y: two elements x E X and y E Y
correspond under G if (x, y) E G. If one returns to the preceding example and
replaces the set H by the set of all men, and does not assume that the society
is monogamous, the relation "y is one of the wives of x" is a correspondence
between Hand F. One does not insist on the existence for every x E H of
ayE F such that (x, y) E G, nor does one insist that such a y be unique;
the x for which there exists such a y constitute the set of definition of the
correspondence (the owners of a harem); the y such that one has (x, y) E G
for at least one x constitute the image or the set of values (the women of a
harem). If X = Y = JR, the relation x 2 +3y2 = 1, whose graph G in the plane
is an ellipse, is a correspondence: its set of definition is the set of x such that
Ixl S; 1, its image the set of y such that Iyl S; 1/V3; this correspondence is
not a function since a real number may have two distinct square roots. The
formula x < y is likewise a correspondence (one more often says "relation"
in cases of this sort) whose graph the reader will have no trouble finding.
In actual practice one often uses other expressions. Instead of saying
let f be a function defined on X with values in Y,
one often says
let f be a map of X into Y
or
consider a map f : X --+ Y.
When f is given by a "formula" one also, for example, speaks of
the map x ~ x 3 of X into Y,
assuming that this makes sense; do not confuse the signs --+ and ~; the
string x ~ x 3 does not denote a map of the set x into the set. x 3 , it denotes
the function or map which to each element x of X associates the element x 3
ofY.
Let us again observe that in mathematics, when speaking of a function
or map f, one must specify the set X on which f is defined and the set Y
in which it takes its values. To speak without further specification of ''the
function x 2 " is meaningless 21 . The map x ~ x 2 of the interval 0 S; x S; 1
21 The logicians nevertheless speak of functional relations without specifying the
sets of departure or arrival: they mean a relation R{ x, y} between two "variables"
x and y such that
R{x,y'} & R{x,y"} implies y' = y".