HW_1_Solutions

HW_1_Solutions - MATH 413 HW 1 Solution to selected...

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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 universal-existential 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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HW_1_Solutions - MATH 413 HW 1 Solution to selected...

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