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/2subsets; 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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 Fall '09
 WILDSTROM
 Natural number, Prime number, positive integer, British B class submarine, B type proanthocyanidin

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