Algebra Know-It-ALL

218 Mappings, Relations, and Functions

can proceed in Fig. 13-8 along the dashed “square spiral,” increasing n by 1 each time we move to the next

block, and letting r be equal to the rational number in the block. When we follow that pattern, the ordered

pairs in the form (n,r) are (0,0), (1,1/2), (2,1), (3,−1/2), (4,−1), (5,1/3), (6,2/3), (7,2/2), (8,2), (9,−1/3),

(10,−2/3), (11,−2/2), (12,−2), and so on forever!

Some of the r values in the range of this relation have more than one n counterpart in the domain. For

example, if r= 1, then n= 2. But when r= 2/2 (which is equal to 1), we have n= 7. The same thing hap-

pens with many other rationals. For example, the ordered pairs (4,−1) and (11,−2/2) have the same value

ofr, because − 1 =−2/2. If we keep writing out the list of ordered pairs for a long time, we’ll keep coming

across “dupes” such as this.

An injection must be one-to-one, but the relation we’ve defined here is not of that sort! Because it’s

not an injection, this relation can’t be a bijection. But it’s a surjection. For every rational number r, we

can always find at least one natural number n that maps to it. Remember that in Chap. 9, we deliberately

engineered this two-dimensional list so it would account for every possible rational number.

Examples of Functions

Afunction is a relation in which every element in the domain has at most one element in the

range. That is, for every value of the independent variable, there can never be more than one

value for the dependent variable. Even so, a single value of the dependent variable might be

mapped from two, three, four, or more values of the independent variable—even infinitely

many. Let’s look at some examples of functions.

Add 1 to the input

Consider a relation in which the independent variable is x and the dependent variable is y,

and for which the domain and range are both the entire set of real numbers. Our relation is

defined as follows:

y=x+ 1

This is a function between x and y, because there’s never more than one value of y for any value

ofx. In fact, for every value of x, there is exactly one value of y. This function is bijective. It

maps values of x onto the entire set R,and it is one-to-one.

Mathematicians name functions by giving them letters of the alphabet such as f, g, and

h. In this notation, the dependent variable is replaced by the function letter followed by the

independent variable in parentheses. We might write

f (x)=x+ 1

to represent the above equation, and then we can say, “f of x equals x plus 1.”

Square the input

Now let’s look at another simple relation. The independent variable is v and the dependent

variable is w. The domain is the entire set of reals, and the range is the set of nonnegative reals.

Here’s the equation that represents the relation:

w=v^2