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Unformatted text preview: MATH 413 HW 1: Solution to selected problems. September 4, 2007 1.1.3.3 The following statement is in universalexistential form. Write the corre sponding statement with the order of quantifiers reversed, and show why it is false. d. There is no largest prime. Solution: Let P be the set of prime natural numbers. The statement above corresponds to ( ∀ p ∈ P )( ∃ q ∈ P )( q > p ) . By reversing the order of the quantifiers we get the statement: ( ∃ q ∈ P )( ∀ p ∈ P )( q > p ) , which asserts that there is a prime number strictly greater than all the prime numbers. This is trivially false, because if it were the case we would get that q > q which is clearly contradictory. Remark: It is easy to get confused here by interpreting the last state ment to be equivalent to the statement: the set of prime numbers is finite. This assertion is false as well but its proof is more elaborated: suppose that P = { p i : 1 ≤ i ≤ N } for some natural number N (i.e  P  = N ). Consider the natural number m = Q N i =1 p i + 1. This number cannot be prime because it is greater than any of the p i ’s. Therefore, it must have a’s....
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 Fall '07
 HUBBARD
 Math, Set Theory, Natural number, Prime number, Cauchy

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