Page 111 - HOW TO PROVE IT: A Structured Approach, Second Edition
P. 111
P1: PIG/
0521861241c03 CB996/Velleman October 20, 2005 2:42 0 521 86124 1 Char Count= 0
Proofs Involving Negations and Conditionals 97
To prove a goal of the form ¬P:
Assume P is true and try to reach a contradiction. Once you have reached
a contradiction, you can conclude that P must be false.
Scratch work
Before using strategy:
Givens Goal
— ¬P
—
After using strategy:
Givens Goal
— Contradiction
—
P
Form of final proof:
Suppose P is true.
[Proof of contradiction goes here.]
Thus, P is false.
2
Example 3.2.2. Prove that if x + y = 13 and y = 4 then x = 3.
Scratch work
The goal is a conditional statement, so according to the first proof strategy in
Section 3.1 we can treat the antecedent as given and make the consequent our
new goal:
Givens Goal
2
x + y = 13 x = 3
y = 4
This proof strategy also suggests what form the final proof should take.
According to the strategy, the proof should look like this:
2
Suppose x + y = 13 and y = 4.
[Proof of x = 3 goes here.]
2
Thus, if x + y = 13 and y = 4 then x = 3.
In other words, the first and last sentences of the final proof have already been
written, and the problem that remains to be solved is to fill in a proof of x = 3
