Page 150 - HOW TO PROVE IT: A Structured Approach, Second Edition

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136 Proofs
3.5. Proofs Involving Disjunctions

Suppose one of your givens in a proof has the form P ∨ Q. This given tells
you that either P or Q is true, but it doesn’t tell you which. Thus, there are two
possibilities that you must take into account. One way to do the proof would
be to consider these two possibilities in turn. In other words, first assume that
P is true and use this assumption to prove your goal. Then assume Q is true
and give another proof that the goal is true. Although you don’t know which of
these assumptions is correct, the given P ∨ Q tells you that one of them must
be correct. Whichever one it is, you have shown that it implies the goal. Thus,
the goal must be true.
The two possibilities that are considered separately in this type of proof –
the possibility that P is true and the possibility that Q is true – are called cases.
The given P ∨ Q justifies the use of these two cases by guaranteeing that these
cases cover all of the possibilities. Mathematicians say in this situation that the
cases are exhaustive. Any proof can be broken into two or more cases at any
time, as long as the cases are exhaustive.

To use a given of the form P ∨ Q:
Break your proof into cases. For case 1, assume that P is true and use this
assumption to prove the goal. For case 2, assume Q is true and give another
proof of the goal.
Scratch work

Before using strategy:
Givens Goal
P ∨ Q —
—
After using strategy:
Case 1: Givens Goal
P —
—
Case 2: Givens Goal
Q —
—
Form of final proof:

Case 1. P is true.
[Proof of goal goes here.]