hw3 - HOMEWORK#3 DUE ON 9/27(1 Let I be a set √ n closed...

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Unformatted text preview: HOMEWORK #3, DUE ON 9/27 (1) Let I be a set √ n closed intervals in R. Show that either there is a disjoint of √ set of at least n intervals in I , or there is a set I ⊆ I such that |I | ≥ n and ∩I = ∅ (there is common point). (2) (a) Show that there exists > 0 such that for all |x| < ∞ 1 2n n √ x. = 1 − 4x n=0 n (b) Prove that for all n positive integer n i=0 2i i 2(n − i) n−i = 4n . (3) Let A be a collection of 13 distinct subsets of {1, 2, 3, . . . , 100} such that for any four sets A1 , A2 , A3 , A4 in A, some Ai ⊂ Aj . Prove that A must contains some five sets B1 , B2 , B3 , B4 , B5 such that B1 ⊂ B2 ⊂ B3 ⊂ B4 ⊂ B5 . Also show that the statement would not be true for |A| = 12. (4) (Exercise 32, Chapter 5) Let S be a set of n elements. Prove that if n is even, then the only antichain of size nn2 of P (S ) is the antichain of all / n/2-subsets; if n is odd, prove that the only antichains of this size are the antichain of all n−1 -subsets and the antichain of n+1 -subsets. (Part of the 2 2 solution was explained in lecture, but please provide the whole proof.) (5) Prove that for n positive integer 4n 2n ≥ . n 2n On the other hand, show that this is not the right asymptotic behavior, i.e. lim n→∞ 2n n 4n 2n 1 = 1. ...
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