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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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This note was uploaded on 02/15/2012 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.
- Fall '10