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# ps1 - (1 f xy = f x f y(2 Claim f x = 0 for all x or f x =...

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18.100B and 18.100C Fall 2011 Problem Set 1 Due September 15th at 4 pm in room 2-108. Hand in parts 1, 2 and 3 separately. Put your name and whether you are registered for 18.100B or 18.100C on each part. Part 1 1. Let m and n be positive integers with no common factor. Prove that if p m/n is rational, then m and n are both perfect squares, that is to say there exist integers p and q such that m = p 2 and n = q 2 . (This is proved in Proposition 9 of Book X of Euclid’s Elements ). 2. Problem 6 from page 22. Part 2 3. Problem 7 from page 22. Part 3 4. Problem 8 from page 22. Students registered for 18.100C should write this problem up in LaTeX. 5. Let R be the set of real numbers and suppose f : R R is a function such that for all real numbers x and y the following two equations hold f ( x + y ) = f ( x ) + f ( y ) , (1)

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Unformatted text preview: , (1) f ( xy ) = f ( x ) f ( y ) . (2) Claim: f ( x ) = 0 for all x or f ( x ) = x for all x . Prove this claim using the following steps: (a) Prove that f (0) = 0 and that f (1) = 0 or 1. (b) Prove that f ( n ) = nf (1) for every integer n and then that f ( n/m ) = ( n/m ) f (1) for all integers n,m such that m 6 = 0. Conclude that either f ( q ) = 0 for all rational numbers q or f ( q ) = q for all rational numbers q . 1 (c) Prove that f is nondecreasing, that is to say that f ( x ) ≥ f ( y ) whenever x ≥ y for any real numbers x and y . (d) Prove that if f (1) = 0 then f ( x ) = 0 for all real numbers x . Prove that if f (1) = 1 then f ( x ) = x for all real numbers x . 2...
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ps1 - (1 f xy = f x f y(2 Claim f x = 0 for all x or f x =...

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