Page 114 - HOW TO PROVE IT: A Structured Approach, Second Edition

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100 Proofs
After using strategy:
Givens Goal
¬P P
—
—

Form of final proof:
[Proof of P goes here.]
Since we already know ¬P, this is a contradiction.

Although we have recommended proof by contradiction for proving goals
of the form ¬P, it can be used for any goal. Usually it’s best to try the other
strategies first if any of them apply; but if you’re stuck, you can try proof by
contradiction in any proof.
The next example illustrates this and also another important rule of proof-
writing: In many cases the logical form of a statement can be discovered
by writing out the definition of some mathematical word or symbol that
occurs in the statement. For this reason, knowing the precise statements of
the definitions of all mathematical terms is extremely important when you’re
writing a proof.

Example 3.2.3. Suppose A, B, and C are sets, A \ B ⊆ C, and x is anything
at all. Prove that if x ∈ A \ C then x ∈ B.

Scratch work
We’re given that A \ B ⊆ C, and our goal is x ∈ A \ C → x ∈ B. Because the
goal is a conditional statement, our first step is to transform the problem by
adding x ∈ A \ C as a second given and making x ∈ B our goal:

Givens Goal
A \ B ⊆ C x ∈ B
x ∈ A \ C
The form of the final proof will therefore be as follows:

Suppose x ∈ A \ C.
[Proof of x ∈ B goes here.]
Thus, if x ∈ A \ C then x ∈ B.

The goal x ∈ B contains no logical connectives, so none of the techniques
we have studied so far apply, and it is not obvious why the goal follows from