Figure 2

If S and T are subsets of R and f is graphed as above, this condition states that horizontal lines intersect G at most once.

Given f: S  T, we say that f maps onto a subset B of T pro­vided that B = Im(f). In particular, we say f maps onto T provided that Im(f) = T. In terms of the graph G of f, f maps S onto T if and only if:

for each y  T there is at least one x  S such that (x, y)  G.

A function f: S  T that is one-to-one and maps onto T is called a one-to-one correspondence between S and T. Thus f is a one-to-one corre­spondence if and only if:

for each y  T there is exactly one x  S such that (x, y)  G.

These three kinds of special functions are illustrated in Figure 3.

Figure 3

Before we turn to mathematical examples, we illustrate the ideas in a nonmathematical setting.

EXAMPLE 2 Suppose that each student in a class S is assigned a seat number

from the set T = {1, 2, ..., 75}. This assignment provides a function f :S  T; thus for each student s, f(s) represents his or her seat number. The func­tion will be one-to-one provided that no two students are assigned the same seat number. In this case, the class cannot have more than 75 students. The function will map S onto T provided that every number in T has been assigned to at least one student. Note that for this to happen the class must have at least 75 students. The only way that f can possibly be a one-to-one correspondence of S onto T is if the class has exactly 75 students.

If we view the function f as a set of ordered pairs, it will consist of pairs in S  T like (Ann Nelson, 73). ■

EXAMPLE 3 (a) We define f: N  N by the rule f(n) = 2n. Then f is one-to-

one since

implies 2n₁ = 2n₂ implies n₁ = n₂.

However, f does not map N onto N since Im(f) consists only of the even natural numbers.

(b) Let  be an alphabet. Then length(w)  N for each word w in ^*. Thus “length” is a function from ^* onto N. [Note that functions can have fancier names than “f.”] To see this, recall that  is nonempty, so  contains some letter, say a. Now 0 = length(), 1 = length(a), 2 = length(aa), etc. The function length is not one-to-one unless S has only one element. ■

EXAMPLE 4 We prove that f: R  R defined by f(x) = 3x - 5 is a one-to-one cor-­

respondence of R onto R. To check that f is one-to-one we need to show that

if f(x)=f(x') then x = x',

i.e.,

if 3x - 5 = 3x' - 5 then x = x'.

But if 3x - 5 = 3x' - 5, then 3x == 3x' [add 5 to both sides] and this im­plies that x = x' [divide both sides by 3].

To show that f maps R onto R we consider an element y in R. We need to find an x in R such that f(x) = y, i.e., 3x — 5 = y. So we solve for x and obtain x = (y + 5)/3. Thus, given y in R, (y + 5)/3 belongs to R and f((y + 5)/3) = 3({y + 5)/3) - 5 = y. This shows that every y in R belongs to Im(f) so that f maps R onto R. ■

Some special functions occur so often that they have special names. Let S be a nonempty set. The identity function l_S on S is the function that maps each element of S to itself:

l_s(x) = x for all x  S.

Thus the identity function is a one-to-one correspondence of S onto S.

A function f: S  T is called a constant function if there is some y₀  T so that f(x) = y₀ for all x  S. The value a constant function takes does not change or vary as x varies over S.

Consider a set S and a subset A of S. The function on S that takes the value 1 at members of A and the value 0 at the other members of S is called the characteristic function of A and is denoted _A [lowercase Greek chi, sub A]. Thus

Note that _A : S  {0,1} is rarely one-to-one and is usually an onto map. In fact, _A maps S onto {0,1} unless A = S or A = . If either A or S\A has at least two members, then _A is not one-to-one.

Now consider functions f: S  T and g: T  U; see Figure 4. We define the composition g ◦ f: S  U by the rule

g ◦ f(x)=g(f{x)) for all x  S.

One might read the left side “g circle f of x” or “g of f of x.” Com­plicated operations that are performed in calculus or on a calculator can be viewed as the composition of simpler functions.