Figure3
Consider a function : S T. If A is a subset of S, we define
f(A)={f(x):x A}.
Thus (A) is the set of images f(x) as x varies over A. We call f(A) the image of the set A under . We are also interested in the inverse image of a set B in T:
f (B)={x S: f(x) B}.
The set f (B) is called the pro-image of the set B under f.
If is invertible, then [Exercise 16] the pre-image of the subset B of T under equals the image of B under -1, i.e., in this case
If is not invertible, it makes no sense to write f -1(y) or f -1(B), of course. Because f -1(B) can't have any meaning unless it means f (B), some people extend the notation and write f -1(B) for what we denote by f (B), even if f is not invertible. Beware!
For y T we write f (y) for the set f ({y}). That is,
f (y)={x S: f(x)=y}.
This set is the pre-image of the element y under f. Note that solving the equation f(x) = y for x is equivalent to finding the set f (y). That is, f (y) is the solution set for the equation f(x) = y. As with equations in algebra, the set f (y) might have one element, several elements, or no elements at all.
EXAMPLE 6 (a) Consider : R R given by (x) = x2. Then
,
which is the solution set of the equation x2 = 4. The pre-image of the set [1, 9] is
([1,9]) = {x R: x2 [1,9]} = {x R: 1 x2 9}
=[-3, -l][l, 3].
Also we have ([-1,0]) = {0} and ([-1,1]) == [-1,1].
(b) Consider the function g: N N N defined by g(m, n) = m2 + n2. Then g(0) = {(0, 0)}, g (1) = {(0,1), (1,0)}, g (2) = {(1,1)}, g (3) = , g (4) == {(0,2), (2,0)}, etc. For instance, we have g (25) = {(0,5), (3,4), (4, 3), (5,0)}. ■
example 7 (a) Let be an alphabet and let L be the length function on *;
L(w) = length(w) for w *. As we noted in Example 3(b) of § 1.3, L maps onto N. For each k N,
L (k) = {w *: L(w) = k} = {w *:length(w) = k}.
Note that the various sets L (k) are disjoint and that their union is *:
L(0) L(l) L(2) … = *.
(b) Consider h: Z {-1,1} where h(n) = (-1)n. Then
h(l) = {n Z: n is even} and h(-l) = {n Z: n is odd} .
These two sets are disjoint and their union is all of Z:
h(l) h(-l) = 2. ■
It is not a fluke that the pre-images of elements cut the domains into slices in these last two examples. We will see that something like this always happens, and we will learn how to exploit the resulting partition into disjoint pieces.
EXERCISES 1.4
1 . Find the inverses of the following functions mapping R into R.
(a) f(x) = 2x + 3 (b) g(x) = x3 - 2
(c) h(x) = (x - 2)3 (d) k{x) = </x + 7
2. Many hand-held calculators have the functions log x, x2, and 1/x.
(a) Specify the domains of these functions.
Which of these functions are inverses of each other?
Which pairs of these functions commute with respect to composition?
(d) Some hand-held calculators also have the functions sin x, cos x and tan x.
If you know a little trigonometry, repeat parts (a), (b) and (c) for these
functions.
3.
Here arc some functions from N
N
to N:
sum(m,
n)
= m
+ n,
prod(m,
n)
= m
n,
max(m,
n)
= max{m,
n},
min(m,
n)
= min{m,
n};
here * denotes multiplication of integers.
(a) Which of these functions map N N onto N?
(b) Show that none of these functions are one-to-one.
(c) For each of these functions F, how big is the set F (4)?
4. Here are some functions mapping P(N) P(N) into P(N): union(A, B) = A B, inter(a, B) = A B and sym(A, B) = A B.
(a) Show that each of these functions maps P(N) P(N) onto P(N).
(b) Show that none of these functions are one-to-one.
(c) For each of these functions F, how big is the set F()? the set F({0})?
5. Here are two “shift functions” mapping N into N: (n) = n + 1 and g(n) = max{0, n - 1} for n N.
(a) Calculate (n) for n = 0, 1, 2, 3, 4, 73.
(b) Calculate g(n) for n = 0, 1, 2, 3, 4, 73.
(c) Show that is one-to-one but does not map N onto N.
(d) Show that g maps N onto N but is not one-to-one.
(e) Show that g ° = lN but that ° g lN.
6. We define : N N and g: N N as follows: (n) = 2n for all n N, g(n) = n/2 if n is even and g(n} = (n - 1)/2 if n is odd.
(a) Calculate g(n) for n = 0, 1, 2, 3, 4, 73.
(b) Show that g ° = lN but that ° g lN.
7. If : S S and f ° f = ls, then f is its own inverse. Show that the following functions are their own inverses.
(a) The function f: (0, ) (0, ) given by f(x) = 1/x.
(b) The function :P (S) P(S) defined by (A) = Ac.
(c) The function g: R R given by g(x) = 1 - x.
(d) The function rev: C C C C [for C a set] defined by rev(x, y) = (y, x).
8.
Let A
be a subset of some set S
and consider the characteristic function A
of
A.
Find
.
9. Let f: S T and g: T U be invertible functions. Show that g ° f is invertible and that (g ° )-l = -1 ° g-1.
10. Let : S T be an invertible function. Show that f -1 is invertible and that
( -1) –1=f.
11. Consider functions : S T and g: T S such that g o = 1S. Nontrivial examples of such pairs of functions appear in Exercises 5 and 6.
(a) Prove that is one-to-one, (b) Prove that g maps T onto S.
12. Consider the function : R R R R defined by
f(x, y) = (x + y, x – y).
(a) Prove that is one-to-one on R R.
(b) Prove that maps R R onto R R.
(c) Find the inverse function -1.
(d) Find the composite functions f ° f -1 and f f.
13. Let : S T.
(a) Show that f(f (B)) B for subsets B of T.
(b) Show that A f (f(A)) for subsets A of S.
(c) Show that (B1 B2) = (B1) (B2) for subsets B1 and B2 of T.
(d) Under what conditions on B does equality hold in part (a)?
14. Let : S T. Prove or disprove. If false, a single example will suffice.
(a) f(A1 A2) = (A1) (A2) for subsets A1 and A2 of S.
(b) (A1\A2) = (A1)\(A2) for subsets A1 and A2 of S.
(c) (A1) = (A2) implies A1 = A2.
15. (a) One can show that if : T U is one-to-one and if g: S T and h: S T satisfy f o g = f o h, then g = h. Give examples of functions , g and h for which o g = o h but g h.
(b) Give examples of functions , g and h for which g ° f = h ° f but g h.
(c) Give a condition on so that g ° f = h ° f implies g = h.
16. Suppose that the function f: S T is invertible and B t.
(a) Show that if y B, then -1(y) (B).
(b) Show that if x f (B), then x f -1(B).
(c) Show that f (B) = f -1(B).