Page 118 - 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

104 Proofs
The goal is also a conditional statement, so we assume the antecedent and
set the consequent as our new goal:
Givens Goal
P → (Q → R) P →¬Q
¬R
We can also now write a little bit of the proof:

Suppose ¬R.
[Proof of P →¬Q goes here.]
Therefore ¬R → (P →¬Q).

We still can’t do anything with the givens, but the goal is another conditional,
so we use the same strategy again:
Givens Goal
P → (Q → R) ¬Q
¬R
P
Now the proof looks like this:

Suppose ¬R.
Suppose P.
[Proof of ¬Q goes here.]
Therefore P →¬Q.
Therefore ¬R → (P →¬Q).
We’ve been watching for our chance to use our first given by applying
either modus ponens or modus tollens, and now we can do it. Since we know
P → (Q → R) and P, by modus ponens we can infer Q → R. Any conclusion
inferred from the givens can be added to the givens column:
Givens Goal
P → (Q → R) ¬Q
¬R
P
Q → R
We also add one more line to the proof:
Suppose ¬R.
Suppose P.
Since P and P → (Q → R), it follows that Q → R.
[Proof of ¬Q goes here.]
Therefore P →¬Q.
Therefore ¬R → (P →¬Q).