and Congruence
3.1 The Idea of a Function
The word function is most commonly used in connection with calculus and its various elaborations, but the idea occurs, often without the word, in nearly all mathematics. In fact, the first two chapters of this book have been full of func-tions, as we shall now see.
(1) In a field F, to every element a there corresponds a unique negative, —a.
Here we have a function
F —> F
a1--> —a
for every a. (We use —> between sets, and 1—> between elements of the sets.) (2) In an ordered field F, to every element x there corresponds a unique
num-ber jxl, called the absolute value of x. The rule of correspondence is that if x _.: 0, then the number corresponding to x is x itself, and if x < 0, then the number corresponding to x is —x. Thus we have a function,
F—* F,
x 1—* Ix' for every x.
under which
under which
3.1 The Idea of a Function 49
(3) Suppose that F is a Euclidean ordered field. Let F+ be the set of all ele-ments of F that are 0. To each element a of F+ there corresponds a unique element VC; of F. (Recall the Euclidean completeness postulate and the definition of "\/7z.) Here we have a function,
F+ ---> F+ , under which
a 1--> Vet for every a in F.
(4) The operation of addition in a field F can be considered as a function, once we have the idea of the product of two sets. For any pair of sets A, B, the product A
x
B is the set of all ordered pairs (a, b), where a E A and b E B.We allow the possibility that A = B. Thus, when we identify a point P of a coordinate plane by giving a pair of coordinates (x,y), we are associ-ating with P an element of the product fIB x FR of the real numbers with themselves.
Consider now the operation of addition in a field E Under this operation, to every pair (a, b) of numbers in F there corresponds a number a + b, called their sum. This can be regarded as a function,
F X F ---> F, where
(a, b)i--> a + b for every (a, b) in F x E
Obviously multiplication can be regarded in the same way.
Note that in these situations there are always three objects involved: first, a set A of objects to which things are going to correspond; second, a set B which contains the objects that correspond to elements of A ; and third, the correspon-dence itself, which associates with every element of A a unique element of B.
The set A is called the domain of definition, or simply the domain. The set B is called the range. The correspondence itself is called the function. In the ex-amples that we have been discussing, these are as follows.
Table 3.1
Domain Range Law
F F F±
F X F
F F F+
F
a 1—> —a a 1—> lal a 1-->
Va
(a, b) i— a + b
50 Distance and Congruence
In the third column, we have described the function by giving the law of correspondence.
A less simple example would be the function which assigns to every positive real number its common logarithm. Here the domain A is the set of all positive real numbers, the range B is the set of all real numbers, and the law of corre-spondence is xi—> log, x. Here the expression log', x is an example of func-tional notation. If the function itself is denoted by f, then f(x) denotes the object corresponding to x. For example, if f is the absolute value function, then
f(1) = 1, f(-1) = 1, f(-5) = 5,
and so on. Similarly, if g is the "positive square root function," then g(4) = 2, g(16) = 4, g(8) = 2g(2) = 2 • "\/,
and so on. We can also use functional notation for addition, if we want to (which we usually don't). If s is the "sum function," then
s(a, b) = a + b , so that
s(2, 3) = 5, s(5, 4) = 9 , and so on. Similarly, if
p
is the "product function," thenp(a, b) = ab , so that
p(5, 4) = 20 and p(7, 5) = 35 .
To sum up, a function f is defined if we describe three things: (1) a set A, called the domain, (2) a set B called the range, and (3) a law of correspondence under which to every element a of A there corresponds a unique element b of B. If a E A, then f(a) denotes the corresponding element of B. We indicate the function f, the domain A, and the range B by writing
f: A .--> B , and we say that f is a function of A into B.
We define composition of functions in the way which is familiar from cal-culus. Thus, given
f: A --> B and
g: B --> C,
the composition g(f) is the function A —> C under which, for every a in A, al—* g(f(a)). For example, if we are using the functions s and
p
to describe sums and products, then3.1 The Idea of a Function 51
means ab + ac, and
means
s(p(a, b),p(a, c))
p(a , s(b, c))
a(b + c).
Finally, we define two special types of function which have special impor-tance. If every b in B is = f (a) for at least one a in A, then we say that f is a func-tion of A onto B. If every b in B is = f(a) for exactly one a in A, then we say that f is a one-to-one correspondence between A and B, and we write
f: A <---> B .
For example, the function f: D --> R, x 1—> x 3 is a one-to-one correspondence.
The function g: D --> R, x 1—* x 2 is not a one-to-one correspondence because, in the range, every positive number appears twice, and no negative number ap-pears at all. The function under which x 1—> —x is a one-to-one correspondence.
(Proof? You need to check that each number y is = —x for exactly one number x.) Similarly, the function x —> 1/x is a one-to-one correspondence; here
A = B = {xlx 0}.
If f is a one-to-one correspondence, then there is a function f -': B <---> A ,
called the inverse of f, which reverses the action of f. That is, f -'(b) = a if f(a) = b. The symbol f -1 is pronounced "f-inverse." When we say that a func-tion has an inverse, this is merely another way of saying that the funcfunc-tion is a one-to-one correspondence.
Given a function
f: A ---> B .
The image of A is the set of all elements of B that appear as values of the func-tion. Thus the image is
{b la E A and b = f(a)}.
In other words, the image is the smallest set that might have been used as the range of the function. For example, if the function
f: [1:R ---> [1:R
is defined by the condition f(x) = x2 for every x, then the range is the set R of all real numbers, and the image is the set of all nonnegative real numbers.
The question may arise why we define functions in such a way as to per-mit the range to be a bigger set than the image. We might have stated the
52 Distance and Congruence
definition in such a way that every function would be onto. But such a defini-tion would be unmanageable. For example, suppose that we define a funcdefini-tion, in calculus, by the equation
f(x) = x4 — 7x3 + 3x2 — 17x + 3 .
This is a function of R into R. To find out what the image is, we would have to find out where this function assumes its minimum; this is a problem in calculus, leading to a difficult problem in algebra. If we required that the image be known for the function to be properly defined, we could not state our calculus problem without first solving it; and this would be an awkward proceeding.
(The definitions given in this section are standard, but in some books on modern algebra, different definitions are given, as follows. Given a function f: A ---> B. B is called the co-domain. If f (a) = f (b) implies a = b, then f is injective.
If f (A) = B —that is, if B is the image—then f is surjective. If both these condi-tions hold, then f is bijective, and f is called a bijection. Thus a bijection is a one-to-one correspondence.)