Page 390 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Summary of Proof Techniques

Note: Paragraphs marked with the symbol D explain how to use each technique
P
in Proof Designer.

To prove a goal of the form:
1. ¬P:
(a) Reexpress as a positive statement.
P D: Select the goal, give the Reexpress command in the Strategy menu,
and use the Reexpress Negative button in the Reexpress dialog box.
(b) Use proof by contradiction; that is, assume that P is true and try to reach
a contradiction.
P D: Select the goal and give the Contradiction command in the Strategy
menu. If you already know which given you are planning to contradict,
you can select it too before giving the Contradiction command, and
Proof Designer will indicate what you have to prove to achieve the
desired contradiction.
2. P → Q:
(a) Assume P is true and prove Q.
P D: Select the goal and give the Direct command in the Strategy menu.
(b) Prove the contrapositive; that is, assume that Q is false and prove that
P is false.
P D: Select the goal and give the Contrapositive command in the Strategy
menu.
3. P ∧ Q:
Prove P and Q separately. In other words, treat this as two separate goals:
P, and Q.
P D: Select the goal and give the Conjunction command in the Infer menu.
If you already have either P or Q as a given, you can select it too,
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