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# ps2 - f is also a surjection(c No ﬁnite set can be in...

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18.100B and 18.100C Fall 2011 Problem Set 2 Due September 22nd 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. Problem 15 from page 23. Students registered for 18.100C should write this problem up in LaTeX. 2. Problem 9 from page 43. Part 2 3. (a) Give an alternate proof that | x · y | ≤ | x || y | for any x, y R n and for any n N by taking the inner product of the vector | x | y - | y | x with itself and using the fact that this is nonnegative. (b) Let a 1 , . . . , a n be positive real numbers. Prove that if ( a 1 + · · · + a n ) 1 a 1 + · · · + 1 a n M for some M > 0, then n M . When does equality hold? 4. Prove that a set is infinite in the sense of § 2.4 if and only if it is in bijec- tion with a proper subset of itself by proving the following statements: (a) If X is a proper subset of J n for some n N , then either X is empty or X is in bijection with J k for some k N , k < n . (b) If f : J k J is an injection, then k . Moreover, k = if and

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Unformatted text preview: f is also a surjection. (c) No ﬁnite set can be in bijection with a proper subset of itself. (d) N is inﬁnite. (e) If X is inﬁnite, there is an injection f : N → X . Finally, use (c), (d), and (e) to conclude. Part 3 1 5. Let S be an ordered set with the property that every nonempty subset of S has both a least upper bound in S and a greatest lower bound in S . (a) Prove that if f : S → S satisﬁes f ( x ) ≤ f ( y ) for any x,y ∈ S with x ≤ y , then there exists an x ∈ S for which f ( x ) = x . (b) Give an example of an inﬁnite subset of Q with this property. (c) Give an example of an uncountable subset of R with this property. 6. Problem 2 from page 43. You may use without proof the fact that if a ,...a n are integers which are not all zero, then there are at most n complex solutions to a z n + a 1 z n-1 + ··· + a n = 0. 2...
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