Page 115 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Proofs Involving Negations and Conditionals 101
the givens. Lacking anything else to do, we try proof by contradiction:
Givens Goal
A \ B ⊆ C Contradiction
x ∈ A \ C
x /∈ B
As before, this transformation of the problem also enables us to fill in a few
more sentences of the proof:
Suppose x ∈ A \ C.
Suppose x /∈ B.
[Proof of contradiction goes here.]
Therefore x ∈ B.
Thus, if x ∈ A \ C then x ∈ B.
Because we’re doing a proof by contradiction and our last given is now a
negated statement, we could try using our strategy for using givens of the form
¬P. Unfortunately, this strategy suggests making x ∈ B our goal, which just
gets us back to where we started. We must look at the other givens to try to find
the contradiction.
In this case, writing out the definition of the second given is the key to
the proof, since this definition also contains a negated statement. By definition,
x ∈ A \ C means x ∈ A and x /∈ C. Replacing this given by its definition gives
us:
Givens Goal
A \ B ⊆ C Contradiction
x ∈ A
x /∈ C
x /∈ B

Now the third given also has the form ¬P, where P is the statement x ∈ C,so
we can apply the strategy for using givens of the form ¬P and make x ∈ C our
goal. Showing that x ∈ C would complete the proof because it would contradict
the given x /∈ C.
Givens Goal
A \ B ⊆ C x ∈ C
x ∈ A
x /∈ C
x /∈ B
Once again, we can add a little more to the proof we are gradually writing
by filling in the fact that we plan to derive our contradiction by proving x ∈ C.