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proofs - Math 220 October 9 2003 To PROVE a statement of...

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Math 220 October 9, 2003 To PROVE a statement of the form .... In each case below, we assume we want to prove a statement of the given form. Some forms can be handled by more than one technique. P Q : Prove both P and Q . • ¬ P : Usually this comes in the form of “prove P is false.” Determine the negation of P and then prove this new statement. P Q : (Direct) Assume that P is true. Under this assumption prove that Q is true. P Q : (Contrapositive) Assume that Q is false. Under this assumption prove that P is false. P Q : (Convert to ) P Q is equivalent to ¬ P Q , so can use techniques for OR statements. P Q : This is the same as ( P Q ) ( Q P ), so prove P Q and then prove Q P . P Q : (Method 1) Use a proof by cases (see below). In each case, either prove P or prove Q . Usually you would do this if your OR statement is combined with a FOR ALL statement: e.g., x, P ( x ) Q ( x ).
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