Page 116 - HOW TO PROVE IT: A Structured Approach, Second Edition
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                                   102                         Proofs
                                   We also add the definition of x ∈ A \ C to the proof, inserting it in what seems
                                   like the most logical place, right after we stated that x ∈ A \ C:

                                       Suppose x ∈ A \ C. This means that x ∈ A and x /∈ C.
                                         Suppose x /∈ B.
                                             [Proof of x ∈ C goes here.]
                                           This contradicts the fact that x /∈ C.
                                         Therefore x ∈ B.
                                       Thus, if x ∈ A \ C then x ∈ B.

                                     We have finally reached a point where the goal follows easily from the
                                   givens. From x ∈ A and x /∈ B we conclude that x ∈ A \ B. Since A \ B ⊆ C
                                   it follows that x ∈ C.

                                   Solution
                                   Theorem. Suppose A, B, and C are sets, A \ B ⊆ C, and x is anything at all.
                                   If x ∈ A \ C then x ∈ B.
                                   Proof. Suppose x ∈ A \ C. This means that x ∈ A and x /∈ C. Suppose x /∈ B.
                                   Then x ∈ A \ B, so since A \ B ⊆ C, x ∈ C. But this contradicts the fact that
                                   x /∈ C. Therefore x ∈ B. Thus, if x ∈ A \ C then x ∈ B.

                                     The strategy we’ve recommended for using givens of the form ¬P only
                                   applies if you are doing a proof by contradiction. For other kinds of proofs,
                                   the next strategy can be used. This strategy is based on the fact that givens of
                                   the form ¬P, like goals of this form, may be easier to work with if they are
                                   reexpressed as positive statements.

                                     To use a given of the form ¬P:
                                       If possible, reexpress this given in some other form.

                                     We have discussed strategies for working with both givens and goals of
                                   the form ¬P, but only strategies for goals of the form P → Q.Wenow fill
                                   this gap by giving two strategies for using givens of the form P → Q.We
                                   said before that many strategies for using givens suggest ways of drawing
                                   inferences from the givens. Such strategies are called rules of inference. Both
                                   of our strategies for using givens of the form P → Q are examples of rules of
                                   inference.


                                     To use a given of the form P → Q:
                                       If you are also given P, or if you can prove that P is true, then you can
                                   use this given to conclude that Q is true. Since it is equivalent to ¬Q →¬P,
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