1873_solutions

# Exists a number w such that f x g x for all numbers x

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Unformatted text preview: at f x gx. c. For every number x there exists a number   0 such that for every number t satisfying the condition |x t| , we have |f x f t | 1. The denial: There exists a number x such that for every number   0 there is at least one number t satisfying the condition |x t |   such that |f x f t | 1. d. There exists a number   0 such that for every pair of numbers x and t satisfying the condition |x t| , we have |f x f t | 1. The denial: For every number   0 it is possible to find a pair of numbers x and t satisfying the condition |x t |   and for which |f x f t | 1. 7 e. For every number  0 and for every number x, there exists a number   0 such that for every number t satisfying |x t| , we have |f t f x | . The denial: There exists a number  0 and a number x such that for every number   0 it is possible to find a number t such that |x t |   and |f x f t | . f. For every positive number there exists a positive number  such that for every pair of numbers x and t satisfying the condition |x t| , we have |f x f t | . The denial: There exists a number  0 such that for every number   0 it is possible to find a pair of numbers x and t satisfying the condition |x t |   for which |f x f t | . 4. Explain why the statement P  Q is equivalent to the statement P Q. The assertion P  Q says that if P is true then Q must also be true. This assertion says nothing at all about Q in the event that P is false. The only way in which the assertion P  Q can be false is that P is true and Q is not. In other words, the denial of the condition P  Q says that P Q. 5. Explain why the statement is false and Q is true). P  Q is equivalent to the assertion that either (P is true and Q is false) or (P The assertion P  Q says that P and Q have the same truth value. So the assertion that they don’t, which means that one of them is true and the other is false. 6. Explain why the statement P Q is equivalent to the statement P P  Q says Q. The assertion P Q says that at least one of the statements P and Q is true. So the denial of the the assertion P Q says that they are both false. 7. Explain why the statement P  Q statements Q and R are false. R is equivalent to the assertion that P is true and that both of the The denial of the assertion P  Q R says that P is true but that the assertion Q R is false. In other words, it says that P is true and that both of the statements Q and R are false. 8. Explain why the converse of the statement P  Q R is equivalent to the condition R  P QP. The converse of the statement P  Q R says that Q R  P and this says that P must be true if at least one of the statments Q and R are true. In other words, the satement Q R  P says that Q  P and also that R  P. 9. Write the assertion P  Q R as simply as you can in its contrapositive form. The contrapositive form of the assertion P  Q R says that Q R  P and this says that if both of the statements Q and R are false then P is false. In other words, this co...
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