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

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E) (P ⇒ Q) ⇒ R and (P ∧ ∼Q) ∨ R. ⇒ Q and (∼P ∨ Q) ∧ (∼Q ∨ P). (f) P ⇐ Give, if possible, an example of a true conditional sentence for which (a) the converse is true. (b) the converse is false. (c) the contrapositive is false. (d) the contrapositive is true. Give, if possible, an example of a false conditional sentence for which (a) the converse is true. (b) the converse is false. (c) the contrapositive is false. (d) the contrapositive is true. Copyright 2011 Cengage Learning, Inc. All Rights Reserved.

E) 5 + 6 = 6 + 5 iff 7 + 1 = 10. (f) A parallelogram has three sides iff 27 is prime. (g) The Eiffel Tower is in Paris if and only if the chemical symbol for helium is√H. √ √ √ √ √ √ √ 10 + 13 < 11 + 12 iff 13 − 12 < 11 − 10. (h) (i) x 2 ≥ 0 iff x ≥ 0. ) (j) x 2 − y 2 = 0 iff (x − y)(x + y) = 0. ) (k) x 2 + y 2 = 50 iff (x + y) 2 = 50. ) 7. Make truth tables for these propositional forms. ⇒ P). (a) P ⇒ (Q ∧ P). ૺ (b) (∼P ⇒ Q) ∨ (Q ⇐ ⇒ P). ૺ (c) ∼Q ⇒ (Q ⇐ (d) (P ∨ Q) ⇒ (P ∧ Q). (e) (P ∧ Q) ∨ (Q ∧ R) ⇒ P ∨ 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. This is not surprising since P ⇒ (Q ⇒ R) is equivalent to (P ∧ Q) ⇒ R.

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