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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.

Ask a homework question - tutors are online