Page 112 - HOW TO PROVE IT: A Structured Approach, Second Edition

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98 Proofs
between these two sentences. The givens–goal list summarizes what we know
and what we have to prove in order to solve this problem.
The goal x = 3 means ¬(x = 3), but because x = 3 has no logical connec-
tives in it, none of the equivalences we know can be used to reexpress this goal
in a positive form. We therefore try proof by contradiction and transform the
problem as follows:
Givens Goal
2
x + y = 13 Contradiction
y = 4
x = 3
Once again, the proof strategy that suggested this transformation also tells
us how to fill in a few more sentences of the final proof. As we indicated earlier,
these sentences go between the first and last sentences of the proof, which were
written before.
2
Suppose x + y = 13 and y = 4.
Suppose x = 3.
[Proof of contradiction goes here.]
Therefore x = 3.
2
Thus, if x + y = 13 and y = 4 then x = 3.
The indenting in this outline of the proof will not be part of the final proof.
We have done it here to make the underlying structure of the proof clear. The
first and last lines go together and indicate that we are proving a conditional
statement by assuming the antecedent and proving the consequent. Between
these lines is a proof of the consequent, x = 3, which we have set off from the
first and last lines by indenting it. This inner proof has the form of a proof by
contradiction, as indicated by its first and last lines. Between these lines we
still need to fill in a proof of a contradiction.
At this point we don’t have a particular statement as our goal; any impossible
conclusion will do. We must therefore look more closely at the givens to see if
some of them contradict others. In this case, the first and third together imply
that y = 4, which contradicts the second.

Solution
2
Theorem. If x + y = 13 and y = 4 then x = 3.
2
Proof. Suppose x + y = 13 and y = 4. Suppose x = 3. Substituting this into
2
the equation x + y = 13, we get 9 + y = 13, so y = 4. But this contradicts
2
the fact that y = 4. Therefore x = 3. Thus, if x + y = 13 and y = 4 then
x = 3.