hw3 - (c) There is a largest integer. (d) There exist...

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Introduction to Mathematical Reasoning Fall 2009 Assignment #3: Due Wednesday, 10/21/09 Part I: 1. For each of the following statements, do the following things: Translate it into symbols. (Be sure that your symbolic statement explicitly in- cludes implied universals and domains of quanti±ers.) Negate the symbolic statement and simplify. (In particular, this means to remove parentheses in expressions of the form ( . . . ).) Translate the negated statement back into a clear and precise English sentence, without using the word “no” or “not.” (a) For every real number x , there is an integer that is greater than x . (b) There is an integer that is greater than every real number.
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Unformatted text preview: (c) There is a largest integer. (d) There exist integers p and q such that q > 0 and p 2 /q 2 = 2. (e) Between any integer and any larger integer, there is a real number. (f) There is an integer that is not the square of any integer. (g) Every real number has a unique cube root. (h) For every R + , there is a positive real number such that | x-2 | < implies | x 2-4 | < whenever x R . 2. Find predicates P ( x ) and Q ( x ) such that one of the following statements is true and the other is false: x R , P ( x ) Q ( x ) and ( x R , P ( x )) ( x R , Q ( x )) ....
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