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4130sols1

# 4130sols1 - 4130 HOMEWORK 1 Due Thursday February 4(1...

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4130 HOMEWORK 1 Due Thursday February 4 (1) Section 1.1.3 Exercise 2b. “The only even prime is 2.” There are many different ways of approaching the problem. One way is n N ( n is even n is prime = n = 2) . The negation is n N ( n is even n is prime n 6 = 2) . That is, “There exists an even prime which is not equal to 2.” (2) Section 1.1.3 Exercise 3b. “Every nonzero rational number has a rational reciprocal.” x Q \ { 0 }∃ y Q ( xy = 1) . The corresponding statement with quantifiers reversed is: y Q x Q \ { 0 } ( xy = 1) . This is false, because if y Q is such that yx = 1 for all x Q \ { 0 } then y = 2 y = 1 which is impossible. (3) Let A be a set and let P ( a ) be a statement about an element of a . We write ! a A P ( a ) for “there exists a unique a A such that P ( a )”. (a) Write the statement ! a A P ( a ) in a form which uses the quantifiers and , and no connec- tives apart from , and ¬ .

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