Let n be a positive integer with n > 20 , and let 
with 
1. Show that S possess two disjoint subsets, the sum of whose elements are equal.
Homework Answers
4. Let n be a positive integer with n > 20, and let S (1,2.. n21 with IS- (a) Show that S possesses two different 3-element subsets, the sums of whose elements are equal b) Show that S possesses t...
4. Let n be a positive integer with n > 20, and let S (1,2.. n21 with IS- (a) Show that S possesses two different 3-element subsets, the sums of whose elements are equal b) Show that S possesses two disjoint subsets, the sums of whose elements are equal.
4. Let n be a positive integer with n > 20, and let S (1,2.. n21 with IS- (a) Show that S possesses two different 3-element subsets, the sums of whose...
,n2} with ISI = n. 4. Let n be a positive integer with n > 20, and let S {1, 2, -I with a) Show that S possesses two dilferent 3-element subsets, the sums of whose elements are equal. (b) Show tha...
,n2} with ISI = n. 4. Let n be a positive integer with n > 20, and let S {1, 2, -I with a) Show that S possesses two dilferent 3-element subsets, the sums of whose elements are equal. (b) Show that S possesses two disjoint subsets, the sums of whose elements are equal
,n2} with ISI = n. 4. Let n be a positive integer with n > 20, and let S {1, 2, -I with a) Show that...
Let n, and let n be a reduced residue. Let r = odd(). Prove that if...
Let n,
and let
n
be a reduced residue. Let r = odd().
Prove that if r = st for positive integers s and t, then
old(t)
= s.
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3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, w...
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, we want to know if there is a sub-collection of k sets S' C S whose union covers all the elements of U. That is, S k, and Us es SU. Prove that this problem is NP-complete. 992 m SES, si
3. (20 pts) Let ụ be a finite set, and let...
Let a and be be in . Show the following. If gcd(a,b)=1, then for every n...
Let a and be be in . Show
the following. If gcd(a,b)=1, then for every n in there
exist x and y in such
that n=ax+by.
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Let be independent, identically distributed random variables with . Let and for , . (a) Show...
Let be independent, identically distributed random variables with . Let and for , . (a) Show that is a martingale. (b) Explain why satisfies the conditions of the martingale convergence theorem (c) Let . Explain why (Hint: there are at least two ways to show this. One is to consider and use the law of large numbers. Another is to note that with probability one does not converge) (d) Use the optional sampling theorem to determine the probability that ever attains...
For , let have an n-dimensional normal distribution . For any , let denote the vector...
For , let have an n-dimensional normal distribution . For any , let denote the vector consisting of the last n-m coordinates of . a. Find the mean vector and variance covariance matrix of b. Show that is a (n-m) dimensional normal random vector. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...
Prove that for every positive real (important: is not necessarily an integer), that . Hint: For...
Prove that for every positive real (important: is not
necessarily an integer), that
.
Hint: For every , the function
is
strictly growing.
We were unable to transcribe this imageWe were unable to transcribe this imagebe(n") (n log, n) > 0 n
Let be a set. Show that the convex hull of , denoted by , is equal to the set
Let be a set. Show that the convex hull of , denoted by , is equal to the set We were unable to transcribe this imageWe were unable to transcribe this imagecvx(S) We were unable to transcribe this image cvx(S)
Consider the following two problems: Bin Packing: Given n items with positive integer sizes s1, s2,...
Consider the following two problems: Bin Packing: Given n items with positive integer sizes s1, s2, . . . , sn, a capacity C for bins and a positive integer k, is it possible to pack the n items using at most k bins? Partition: Given a set S of n integers, is it possible to partition S into two subsets S1 and S2 so that the sum of the integers in S1 is equal to the sum of the...
4. Let n be a positive integer with n > 20, and let S (1,2.. n21 with IS- (a) Show that S possesses two different 3-element subsets, the sums of whose elements are equal b) Show that S possesses two disjoint subsets, the sums of whose elements are equal.
4. Let n be a positive integer with n > 20, and let S (1,2.. n21 with IS- (a) Show that S possesses two different 3-element subsets, the sums of whose...
,n2} with ISI = n. 4. Let n be a positive integer with n > 20, and let S {1, 2, -I with a) Show that S possesses two dilferent 3-element subsets, the sums of whose elements are equal. (b) Show that S possesses two disjoint subsets, the sums of whose elements are equal
,n2} with ISI = n. 4. Let n be a positive integer with n > 20, and let S {1, 2, -I with a) Show that...
Let n,
and let
n
be a reduced residue. Let r = odd().
Prove that if r = st for positive integers s and t, then
old(t)
= s.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, we want to know if there is a sub-collection of k sets S' C S whose union covers all the elements of U. That is, S k, and Us es SU. Prove that this problem is NP-complete. 992 m SES, si
3. (20 pts) Let ụ be a finite set, and let...
Let a and be be in . Show
the following. If gcd(a,b)=1, then for every n in there
exist x and y in such
that n=ax+by.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Prove that for every positive real (important: is not
necessarily an integer), that
.
Hint: For every , the function
is
strictly growing.
We were unable to transcribe this imageWe were unable to transcribe this imagebe(n") (n log, n) > 0 n
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