Page 145 - HOW TO PROVE IT: A Structured Approach, Second Edition

P1: PIG/
0521861241c03 CB996/Velleman October 20, 2005 2:42 0 521 86124 1 Char Count= 0

Proofs Involving Conjunctions and Biconditionals 131
might try to shorten the proof by writing it as a string of equivalences, starting
with x ∈ A ∩ (B \ C) and ending with x ∈ (A ∩ B) \ C. In this case, if we
start with x ∈ A ∩ (B \ C) and follow the first string of equivalences displayed
above, we come to a statement that is the same as the last statement in the
second string of equivalences. We can then continue by following the second
string of equivalences backward, ending with x ∈ (A ∩ B) \ C.
Solution

Theorem. Suppose A, B, and C are sets. Then A ∩ (B \ C) = (A ∩ B) \ C.
Proof. Let x be arbitrary. Then

x ∈ A ∩ (B \ C)iff x ∈ A ∧ x ∈ B \ C
iff x ∈ A ∧ x ∈ B ∧ x /∈ C
iff x ∈ (A ∩ B) ∧ x /∈ C
iff x ∈ (A ∩ B) \ C.

Thus, ∀x(x ∈ A ∩ (B \C) ↔ x ∈ (A ∩ B)\C), so A ∩ (B \C) = (A ∩ B)\C.

The technique of figuring out a sequence of equivalences in one order and
then writing it in the reverse order is used quite often in proofs. The order in
which the steps should be written in the final proof is determined by our rule
that an assertion should never be made until it can be justified. In particular,
if you are trying to prove P ↔ Q, it is wrong to start your write-up of the
proof with the unjustified statement P ↔ Q and then work out the meanings
of the two sides P and Q, showing that they are the same. You should instead
start with equivalences you can justify and string them together to produce a
justification of the goal P ↔ Q before you assert this goal. A similar technique
can sometimes be used to figure out proofs of equations, as the next example
shows.

Example 3.4.5. Prove that for any real numbers a and b,

2 2
(a + b) − 4(a − b) = (3b − a)(3a − b).

Scratch work
2
2
The goal has the form ∀a∀b((a + b) − 4(a − b) = (3b − a)(3a − b)), so we
start by letting a and b be arbitrary real numbers and try to prove the equation.