By Smith D., Eggen M., Andre R.
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All triangles) (e) All people are honest or no one is honest. (All people) (f) Some people are honest and some people are not honest. (All people) (g) Every nonzero real number is positive or negative. (Real numbers) ૺ (h) Every integer is greater than −4 or less than 6. (Real numbers) (i) Every integer is greater than some integer. (Integers) ૺ (j) No integer is greater than every other integer. (Integers) (k) Between any integer and any larger integer, there is a real number. (Real numbers) ૺ (l) There is a smallest positive integer.
P is equivalent to Q. P is necessary and sufficient for Q. |t | |t| |t| |t| = 2 if and only if t2 = 4. = 2 if, but only if, t 2 = 4. = 2 is equivalent to t 2 = 4. = 2 is necessary and sufficient for t 2 = 4. The word unless is one of those connective words in English that poses special problems because it has so many different interpretations. See Exercise 11. Examples. In these sentence translations, we assume that S, G, and e have been specified. It is not necessary to know the meanings of all the words because the form of the sentence is sufficient to determine the correct translation.
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.