24
I - Sets and Functions
If F c X X Y and G c Y X Z are the graphs of f and g, the graph HeX X Z
of h is the set of ordered pairs (x, z) having the following property: there exists
ayE Y such that both (x, y) E F and (y, z) E G. It is immediate that H is
a graph. The composed map h is denoted go f: thus
(6.1)
(g 0 f)(x) = gl!(x)],
but in fact one always writes 9 0 f (x) instead of (g 0 f) (x). This concept generalises what one does when speaking of the function sin cos x - one ought to
write sin 0 cos - or, in geometry, when defining the "product" of two homotheties, translations, etc.
Given a map f : X -----+ Y, one frequently has to consider those x E X
such that f(x) = b is a given element of Y. As regards their existence, all
cases are clearly possible. The simplest is that where the equation f(x) = b
has at least one solution no matter what b E Yj f is then said to be surjective
(or is a surjection). The map x !-------+ x 3 of IR into IR is surjective, since every
real number, whatever its sign, has a cube root. The map x !-------+ x 2 is not,
because only positive numbers have square roots.
More generally, if one replaces X by one of its subsets A one is led to
introduce the set BeY of y such that the equation f(x) = b has at least
one solution in A (and possibly elsewhere 23 ). This set, denoted by f(A), is
called the image of A by f, a concept familiar from elementary geometry: the
image of a circle by a translation is a circle. Clearly
(6.2)
f(A U B) = f(A) U f(B),
but
the relation f(A n B) = f(A) n f(B) is false
in general, since for bE f(A) n f(B) the equation f(x) = b has at least one
solution in A and at least one solution in B, but why should they be the
same?
The situation is simpler if f is injective or is an injection, i.e. if the equation f(x) = b has at most one solution for every bEY. In this case, it is
clear, one always has f(A n B) = f(A) n f(B).
Together with the concept of image - one also says the direct image- we
have, in the inverse sense, the inverse image under f of a subset B of Y: this
is the set of x E X such that f(x) E Bj the notation is f-l(B). This time
one has
(6.3)
f- 1 (B' U B")
f- 1 (B' n B")
f-l(B') U f- 1 (B"),
f-l(B') n f-l(B")j
23 In mathematics one says what one says and does not say what one does not
say. Observing this rule in public or private life might eliminate a great number
of stupid discussions of the type "You say that the French are racists. Do you
believe that the Americans are not?"
I - Sets and Functions
If F c X X Y and G c Y X Z are the graphs of f and g, the graph HeX X Z
of h is the set of ordered pairs (x, z) having the following property: there exists
ayE Y such that both (x, y) E F and (y, z) E G. It is immediate that H is
a graph. The composed map h is denoted go f: thus
(6.1)
(g 0 f)(x) = gl!(x)],
but in fact one always writes 9 0 f (x) instead of (g 0 f) (x). This concept generalises what one does when speaking of the function sin cos x - one ought to
write sin 0 cos - or, in geometry, when defining the "product" of two homotheties, translations, etc.
Given a map f : X -----+ Y, one frequently has to consider those x E X
such that f(x) = b is a given element of Y. As regards their existence, all
cases are clearly possible. The simplest is that where the equation f(x) = b
has at least one solution no matter what b E Yj f is then said to be surjective
(or is a surjection). The map x !-------+ x 3 of IR into IR is surjective, since every
real number, whatever its sign, has a cube root. The map x !-------+ x 2 is not,
because only positive numbers have square roots.
More generally, if one replaces X by one of its subsets A one is led to
introduce the set BeY of y such that the equation f(x) = b has at least
one solution in A (and possibly elsewhere 23 ). This set, denoted by f(A), is
called the image of A by f, a concept familiar from elementary geometry: the
image of a circle by a translation is a circle. Clearly
(6.2)
f(A U B) = f(A) U f(B),
but
the relation f(A n B) = f(A) n f(B) is false
in general, since for bE f(A) n f(B) the equation f(x) = b has at least one
solution in A and at least one solution in B, but why should they be the
same?
The situation is simpler if f is injective or is an injection, i.e. if the equation f(x) = b has at most one solution for every bEY. In this case, it is
clear, one always has f(A n B) = f(A) n f(B).
Together with the concept of image - one also says the direct image- we
have, in the inverse sense, the inverse image under f of a subset B of Y: this
is the set of x E X such that f(x) E Bj the notation is f-l(B). This time
one has
(6.3)
f- 1 (B' U B")
f- 1 (B' n B")
f-l(B') U f- 1 (B"),
f-l(B') n f-l(B")j
23 In mathematics one says what one says and does not say what one does not
say. Observing this rule in public or private life might eliminate a great number
of stupid discussions of the type "You say that the French are racists. Do you
believe that the Americans are not?"