Page 391 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Summary of Proof Techniques 377
and Proof Designer will only ask you to prove the statement that you
don’t already know.
4. P ∨ Q:
(a) Assume P is false and prove Q, or assume Q is false and prove P.
P D: Select the goal and give the Disjunction command in the Strategy
menu. Proof Designer will ask which statement you are planning to
prove.
(b) Use proof by cases. In each case, either prove P or prove Q.
P D: Use the Cases command in the Strategy menu to break your proof
into cases. Your goal in each case will be P ∨ Q. In each case, select
this goal and give the Disjunction command in the Strategy menu,
and Proof Designer will ask you which statement you plan to prove
in that case. If you don’t want to assume the negation of the other
statement, remove the check mark from the “Assume negations of
others” check box by clicking on it.
5. P ↔ Q:
Prove P → Q and Q → P, using the methods listed under part 2.
P D: Select the goal and give the Biconditional command in the Strategy
menu.
6. ∀xP(x):
Let x stand for an arbitrary object, and prove P(x). (If the letter x already
stands for something in the proof, you will have to use a different letter for
the arbitrary object.)
P D: Select the goal and give the Arbitrary Object command in the Strategy
menu.
7. ∃xP(x):
Find a value of x that makes P(x) true. Prove P(x) for this value of x.
P D: Select the goal and give the Existence command in the Strategy
menu. Proof Designer will ask you what value you want to use for
x. If you’re not sure what to use for x, you can choose a variable to
stand for this value, and fill in the choice of a value for that variable
later.
8. ∃!xP(x):
(a) Prove ∃xP(x) (existence) and ∀y∀z((P(y) ∧ P(z)) → y = z) (unique-
ness).
P D: Select the goal and give the Existence & Uniqueness command in
the Strategy menu.
(b) Prove the equivalent statement ∃x(P(x) ∧∀y(P(y) → y = x)).
P D: Select the goal, give the Reexpress command in the Strategy menu,
and click on the Apply Definition button in the Reexpress dialog box.