Page 183 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Ordered Pairs and Cartesian Products 169
truth set of P(x, y) is the subset of A × B consisting of those assignments that
make the statement come out true. In other words,

truth set of P(x, y) ={(a, b) ∈ A × B | P(a, b)}.

Example 4.1.6. What are the truth sets of the following statements?
1. “x has y children,” where x ranges over the set P of all people and y ranges
over N.
2. “x is located in y,” where x ranges over the set C of all cities and y ranges
over the set N of all countries.
3. “y = 2x − 3,” where x and y range over R.
Solutions

1. {(p, n) ∈ P × N | the person p has n children} = {(Prince Charles, 2), ...}.
2. {(c, n) ∈ C × N | the city c is located in the country n} = {(New York,
United States), (Tokyo, Japan), (Paris, France), ...}.
3. {(x, y) ∈ R × R | y = 2x − 3}={(0, −3), (1, −1), (2, 1),...}. You are
probably already familiar with the fact that the ordered pairs in this set
are the coordinates of points in the plane that lie along a certain straight
line, called the graph of the equation y = 2x − 3. Thus, you can think of
the graph of the equation as a picture of its truth set!

Many of the facts about truth sets for statements with one free variable that
we discussed in Chapter 1 carry over to truth sets for statements with two free
variables. For example, suppose T is the truth set of a statement P(x, y), where
x ranges over some set A and y ranges over B. Then for any a ∈ A and b ∈ B
the statement (a, b) ∈ T means the same thing as P(a, b). Also, if P(x, y)is
true for every x ∈ A and y ∈ B, then T = A × B, and if P(x, y) is false for
every x ∈ A and y ∈ B, then T = ∅.If S is the truth set of another statement
Q(x, y), then the truth set of the statement P(x, y) ∧ Q(x, y)is T ∩ S, and the
truth set of P(x, y) ∨ Q(x, y)is T ∪ S.
Although we’ll be concentrating on ordered pairs for the rest of this chapter,
it is possible to work with ordered triples, ordered quadruples, and so on. These
might be used to talk about truth sets for statements containing three or more
free variables. For example, let L(x, y, z) be the statement “x has lived in y for
z years,” where x ranges over the set P of all people, y ranges over the set C
of all cities, and z ranges over N. Then the assignments of values to the free
variables that make sense in this statement would be ordered triples (p, c, n),
where p is a person, c is a city, and n is a natural number. The set of all such
ordered triples would be written P × C × N, and the truth set of the statement