1873_solutions

Have red hair c if a triangle abc has a right angle

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Unformatted text preview: BC satisfying the the condition AB 2  AC 2  BC 2 . must have a right angle at C. The contrapositive: Every triangle ABC that fails to satisfy the the condition AB 2  AC 2  BC 2 . will not have a right angle at C. The denial: There is at least one triangle ABC that has a right angle at C and for which the condition AB 2  AC 2  BC 2 . does not hold. d. If some cats scratch, then all dogs bite. The converse: If all dogs bite then some cats scratch. The contrapositive: If there exists a dog that does not bite then no cats scratch. The denial: Some cats scratch but not all dogs bite. e. It is with regret that I inform you that someone in this room is smoking. The converse: This statement doesn’t have a converse. The contrapositive: This statement doesn’t have a contrapositive form. The denial: It is without regret that I inform you that someone in this room is smoking. 6 f. If a function is differentiable at a given number then it must be continuous at that number. The converse: If a function is continuous at a given number then it must be differentiable at that number. The contrapositive: If a function is not continuous at a given number then it cannot be differentiable at that number. The denial: There exists a function and a real number such that the function is differentiable at the number but fails to be continuous at that number. g. Every boy or girl alive is either a little liberal or else a conservative. The converse: Every little liberal and every conservative is a living boy or girl The contrapositive: Any individual who fails to be either a little liberal or a conservative cannot be either a living boy or a living girl. The denial: There is at least one living boy or girl who is neither a little liberal nor a conservative. 2. In each of the following exercises, write down a denial of the given statement. a. All cats scratch and some dogs bite. The denial: Either there is a cat that does not scratch or no dogs bite. b. Either some cats scratch or if all dogs bite then some birds sing. The denial: No cats scratch and all dogs bite and no birds sing. c. He walked into my office this morning, told me a pack of lies and punched me on the nose. The denial: Either he did not walk into my office this morning or he did not tell me a pack of lies or he did not punch me on the nose. d. No one has ever seen an Englishman who is not carrying an umbrella. The denial: At least one person has seen an Englishman who is not carrying an umbrella. e. For every number x there exists a number y such that y  x. The denial: There exists a number x such that for every number y we have y x. 3. In each of the following exercises we assume that f and g are given functions. Write down a denial of each of the following statements: a. Whenever x  50, we have f x  g x . The denial: There exists a number x  50 such that f x gx. b. There exists a number w such that f x  g x for all numbers x  w. The denial: For every number w there exists a number x  w such th...
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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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