Page 392 - HOW TO PROVE IT: A Structured Approach, Second Edition

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378 Summary of Proof Techniques
9. ∀n ∈ NP(n):
(a) Mathematical Induction: Prove P(0) (base case) and ∀n ∈ N(P(n) →
P(n + 1)) (induction step).
(b) Strong Induction: Prove ∀n ∈ N[(∀k < nP(k)) → P(n)].

To use a given of the form:

1. ¬P:
(a) Reexpress as a positive statement.
P D: Select the given, give the Reexpress command in the Strategy menu,
and use the Reexpress Negative button in the Reexpress dialog box.
(b) In a proof by contradiction, you can reach a contradiction by proving P.
P D: Select the given and give the Contradiction command in the Strategy
menu.
2. P → Q:
(a) If you are also given P, or you can prove that P is true, then you can
conclude that Q is true.
P D: Select the givens P and P → Q and give the Modus Ponens com-
mand in the Infer menu, and Proof Designer will infer Q. (If you don’t
already have P as a given but you think you can prove it, you can use
the Insert command in the Edit menu to insert a proof of P.)
(b) Use the contrapositive: If you are given or can prove that Q is false,
then you can conclude that P is false.
P D: Select the givens ¬Q and P → Q and give the Modus Tollens com-
mand in the Infer menu, and Proof Designer will infer ¬P.
3. P ∧ Q:
Treat this as two givens: P, and Q.
P D: Select the given and give the Split Up command in the Infer menu.
4. P ∨ Q:
(a) Use proof by cases. In case 1 assume that P is true, and in case 2 assume
that Q is true.
P D: Select the given and give the Cases command in the Strategy menu.
(b) If you are also given that P is false, or you can prove that P is false,
then you can conclude that Q is true. Similarly, if you know that Q is
false then you can conclude that P is true.
P D: Select the givens ¬P (or ¬Q) and P ∨ Q and give the Disjunctive
Syllogism command in the Infer menu.
5. P ↔ Q:
Treat this as two givens: P → Q, and Q → P.
P D: Select the given and give the Split Up command in the Infer menu.