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

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With part (a), for example, we reason as follows: P ⇒ Q is false exactly when P is true and Q is false, which happens exactly when both ∼P and Q are false. Since this happens exactly when ∼P ∨ Q is false, the truth tables for P ⇒ Q and ∼P ∨ Q are identical. 2 are related. 1(a), to P ∧ ∼Q. Recognizing the structure of a sentence and translating the sentence into symbolic form using logical connectives are aids in determining its truth or falsity. The translation of sentences into propositional symbols is sometimes very complicated because some natural languages such as English are rich and powerful with many nuances.

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. This is not surprising since P ⇒ (Q ⇒ R) is equivalent to (P ∧ Q) ⇒ R. The main lesson to be learned from this discussion is that the method of proof you choose will depend on the form of the statement to be proved. The outlines we have given are the most natural, but not the only ways, to construct correct proofs.

C) ∼Q ⇒ (Q ⇐ (d) (P ∨ Q) ⇒ (P ∧ Q). (e) (P ∧ Q) ∨ (Q ∧ R) ⇒ P ∨ R. (f) [(Q ⇒ S) ∧ ( Q ⇒ R)] ⇒ [(P ∨ Q) ⇒ ( S ∨ R)]. Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 2 Conditionals and Biconditionals 17 8. 2 by constructing truth tables for each equivalence. 9. Determine whether each statement qualifies as a definition. (a) y = f (x) is a linear function when its graph is a straight line. (b) y = f (x) is a quadratic function when it contains an x 2 term.