Page 182 - HOW TO PROVE IT: A Structured Approach, Second Edition

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168 Relations
so by one of our strategies for disjunctions from Chapter 3 we can assume
¬(A = ∅∨B = ∅) and prove A = B. Note that by one of DeMorgan’s laws,
¬(A = ∅∨B = ∅) is equivalent to A = ∅∧B = ∅, so we treat this as two
assumptions, A = ∅ and B = ∅. Of course we could also have proceeded dif-
ferently, for example by assuming A = B and B =∅ and then proving A =∅.
But recall from the commentary on part 5 of Theorem 4.1.3 that A = ∅ and
B = ∅ are actually negative statements, so because it is generally better to
work with positive than negative statements, we’re better off negating both of
them to get the assumptions A = ∅ and B = ∅ and then proving the pos-
itive statement A = B. The assumptions A = ∅ and B = ∅ are existential
statements, so they are used in the proof to justify the introduction of y and z.
The proof that A = B proceeds in the obvious way, by introducing an arbitrary
object x and then proving x ∈ A ↔ x ∈ B.
For the ← direction of the proof, we have A = ∅∨B = ∅∨A = B as a
given, so it is natural to use proof by cases. In each case, the goal is easy to
prove.
This theorem is a better illustration of how mathematics is really done than
most of the examples we’ve seen so far. Usually when you’re trying to find
the answer to a mathematical question you won’t know in advance what the
answer is going to be. You might be able to take a guess at the answer and you
might have an idea for how the proof might go, but your guess might be wrong
and your idea for the proof might be flawed. It is only by turning your idea into
a formal proof, according to the rules in Chapter 3, that you can be sure your
answer is right. Often in the course of trying to construct a formal proof you will
discover a flaw in your reasoning, as we did earlier, and you may have to revise
your ideas to overcome the flaw. The final theorem and proof are often the result
of repeated mistakes and corrections. Of course, when mathematicians write
up their theorems and proofs, they follow our rule that proofs are for justifying
theorems, not for explaining thought processes, and so they don’t describe
all the mistakes they made. But just because mathematicians don’t explain
their mistakes in their proofs, you shouldn’t be fooled into thinking they don’t
make any!
Now that we know how to use ordered pairs and Cartesian products to talk
about assigning values to free variables, we’re ready to define truth sets for
statements containing two free variables.

Definition 4.1.5. Suppose P(x, y) is a statement with two free variables in
which x ranges over a set A and y ranges over another set B. Then A × B is the
set of all assignments to x and y that make sense in the statement P(x, y). The