By Smith D., Eggen M., Andre R.

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Assume P. . Therefore R. Case 2. Assume Q. . Therefore R. This method is valid because of the tautology [(P ∨ Q) ⇒ R] ⇐ ⇒ [(P ⇒ R) ∧ (Q ⇒ R)]. ” The two similar statement forms (P ⇒ Q) ⇒ R and P Q (Q Q R) have remarkably dissimilar direct proof outlines. For (P ⇒ Q) ⇒ R, we assume P ⇒ Q and deduce R. We cannot assume P; we must assume P ⇒ Q. On the other hand, in a direct proof of P ⇒ (Q ⇒ R), we do assume P and show Q ⇒ R. Furthermore, after the assumption of P, a direct proof of Q ⇒ R begins by assuming Q is true as well.

D) the contrapositive is true. Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. qxd 18 CHAPTER 1 4/22/10 1:42 AM Page 18 Logic and Proofs 15. Give the converse and contrapositive of each sentence of Exercises 10(a), (b), (c), and (d). Tell whether each converse and contrapositive is true or false. 16. Determine whether each of the following is a tautology, a contradiction, or neither. ૺ (a) [(P ⇒ Q) ⇒ P] ⇒ P. ⇒ P ∧ (P ∨ Q).

The statement of an assumption generally takes the form “Assume P” to alert the reader that the statement is not derived from a previous step or steps. We must be careful about making assumptions, because we can only be certain that what we proved will be true when all the assumptions are true. The most common assumptions are hypotheses given as components in the statement of the theorem to be proved. We will discuss assumptions in more detail later in this section. The statement of an axiom is usually easily identified as such by the reader because it is a statement about a very fundamental fact assumed about the theory.