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# Have been days on which it has rained and on which

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Unformatted text preview: told me that it has never rained on a day on which you have remembered to take your umbrella. Solution: You have not told me that it has never rained on a day on which you have remembered to take your umbrella. 12. You lied when you told me that it has never rained on a day on which you have remembered to take your umbrella. Solution: You did not lie when you told me that it has never rained on a day on which you have remembered to take your umbrella. 13. I was joking when I said that you lied when you told me that it has never rained on a day on which you have remembered to take your umbrella. Solution: I was not joking when I said that you lied when you told me that it has never rained on a day on which you have remembered to take your umbrella. 14. This, Watson, if I mistake not, is our client now. Solution: This statement is meaningless because it refers to itself. We should have been able to expect better logical precision from Sherlock Holmes. 15. a. For every real number x there exists a real number y such that 5y  4x 2x 2  xy y 2 2   . 3x y 3 x 2  xy  y 2 x3 y3 Is this statement is true? Solution: Yes this statement is true. Negation: There exists a real number x such that for every real number y, the equation 5y  4x 2x 2  xy y 2 2   3x y 3 x 2  xy  y 2 x3 y3 is either false or meaningless. b. There exists a real number x such that for every real number y we have 5y  4x 2x 2  xy y 2 2   . 3 3 2  xy  y 2 3x y 3x x y Is this statement is true? Solution: This statement is false because the equation shown her can’t be true when y  x. Negation: For every real number x there exists a real number y for which the equation 5y  4x 2x 2  xy y 2 2   3x y 3 x 2  xy  y 2 x3 y3 is either false or meaningless. c. For every real number x and every real number y x we have 5y  4x 2x 2  xy y 2 2   . 3 3 2  xy  y 2 3x y 3x x y Is this statement is true? 5 Solution: This statement is true. Negation: There exists a real number x and there exists a real number y 5y  4x 2x 2  xy y 2 2   3x y 3 x 2  xy  y 2 x3 y3 x such that the equation is either false or meaningless. Some Exercises on The Use of Conditionals In the exercises that follow you should assume that P, Q, R and S are given statements that may be either true or false. 1. Write down the denial, the converse and the contrapositive form of each of the following statements: a. All cats scratch. The converse: All things that scratch are cats. The contrapositive: All things that do not scratch are not cats. The denial: There is at least one cat that does not scratch. b. If what you said yesterday is correct, then Jim has red hair. The converse: If Jim has red hair then what you said yesterday is correct. The contrapositive: If Jim does not have red hair then what you said yesterday is false. The denial: What you said yesterday is correct and Jim does not have red hair. c. If a triangle ABC has a right angle at C then AB 2  AC 2  BC 2 . The converse: Every triangle A...
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## This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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