Page 393 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Summary of Proof Techniques 379
6. ∀xP(x):
You can plug in any value, say a, for x, and conclude that P(a) is true.
P D: Select the given and give the Universal Instantiation command in the
Infer menu. Proof Designer will ask you what you want to plug in for
x. As with proofs of goals of the form ∃xP(x), if you’re not sure what
to plug in for x, you can choose a variable to stand for the object to
be plugged in, and fill in the choice of a value for that variable later.
7. ∃xP(x):
Introduce a new variable, say x 0 , into the proof, to stand for a particular
object for which P(x 0 ) is true.
P D: Select the given and give the Existential Instantiation command in
the Infer menu.
8. ∃!xP(x):
Introduceanewvariable,say x 0 ,intotheproof,tostandforaparticularobject
for which P(x 0 ) is true. You may also assume that ∀y(P(y) → y = x 0 ).
P D: Select the given and give the Existential Instantiation command in
the Infer menu.

Techniques that can be used in any proof:
1. Proof by contradiction: Assume the goal is false and derive 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.
2. Proofbycases:Considerseveralcasesthatareexhaustive,thatis,thatinclude
all the possibilities. Prove the goal in each case.
P D: If you select a given of the form P ∨ Q and give the Cases command
in the Strategy menu, then Proof Designer will break the proof into
the cases determined by this given. If you select a goal and give the
Cases command, then Proof Designer will ask you to type in some
statement P that will be used to distinguish the cases. In case 1, Proof
Designer will assume that P is true, and in case 2 it will assume that
P is false.