Page 177 - HOW TO PROVE IT: A Structured Approach, Second Edition

P1: PIG/ P2: OYK/
0521861241c04 CB996/Velleman October 20, 2005 2:54 0 521 86124 1 Char Count= 0

4

Relations

4.1. Ordered Pairs and Cartesian Products

In Chapter 1 we discussed truth sets for statements containing a single free
variable. In this chapter we extend this idea to include statements with more
than one free variable.
For example, suppose P(x, y) is a statement with two free variables x and y.
Wecan’tspeakofthisstatementasbeingtrueorfalseuntilwehavespecifiedtwo
values – one for x and one for y. Thus, if we want the truth set to identify which
assignments of values to free variables make the statement come out true, then
the truth set will have to contain not individual values, but pairs of values. We
will specify a pair of values by writing the two values in parentheses separated
by a comma. For example, let D(x, y) mean “x divides y.” Then D(6, 18) is
true, since 6 | 18, so the pair of values (6, 18) is an assignment of values to the
variables x and y that makes the statement D(x, y) come out true. Note that 18
does not divide 6, so the pair of values (18, 6) makes the statement D(x, y) false.
We must therefore distinguish between the pairs (18, 6) and (6, 18). Because
the order of the values in the pair makes a difference, we will refer to a pair
(a, b)asan ordered pair, with first coordinate a and second coordinate b.
You have probably seen ordered pairs before when studying points in the
xy plane. The use of x and y coordinates to identify points in the plane works
by assigning to each point in the plane an ordered pair, whose coordinates
are the x and y coordinates of the point. The pairs must be ordered because,
for example, the points (2, 5) and (5, 2) are different points in the plane. In
this case the coordinates of the ordered pairs are real numbers, but ordered
pairs can have anything at all as their coordinates. For example, suppose we
let C(x, y) stand for the statement “x has y children.” In this statement the
variable x ranges over the set of all people, and y ranges over the set of all
natural numbers. Thus, the only ordered pairs it makes sense to consider when
163