Let A be the set of binary strings containing exactly two 1s. Let B be the set Question 3. of binary strings containing exactly three 1s. Let w be the weight function that returns the length of a binary string. (a) Write down the generating series A(x) and p(x) for A and B respectively with respect to weight w, in closed form (i.e. without using E or +… ·+). (b) Let +(r) = &a(x)+B(x). Express [r"]#(x) as a function of n, for every nonnegative integer n. Using a bijective proof to prove that |S| is equal to the expression you had in part (b), (c) where S = {(a, 3) : a € A, ß E B,\a| + |3| = n}, where Ja| denotes the length of a.
Let A be the set of binary strings containing exactly two 1s. Let B be the set Question 3. of binary strings containing exactly three 1s. Let w be the weight function that returns the length of a binary string. (a) Write down the generating series A(x) and p(x) for A and B respectively with respect to weight w, in closed form (i.e. without using E or +… ·+). (b) Let +(r) = &a(x)+B(x). Express [r"]#(x) as a function of n, for every nonnegative integer n. Using a bijective proof to prove that |S| is equal to the expression you had in part (b), (c) where S = {(a, 3) : a € A, ß E B,\a| + |3| = n}, where Ja| denotes the length of a.
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
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![Question 3.
of binary strings containing exactly three 1s. Let w be the weight function that returns the length of a
binary string.
Let A be the set of binary strings containing exactly two 1s. Let B be the set
(a)
Write down the generating series PA(x) and PB(x) for A and B respectively with respect
to weight w, in closed form (i.e. without using E or +·+).
(b)
Let +(x) = PA(x)ÞB(x). Express [x"]¤(x) as a function of n, for every nonnegative integer
п.
Using a bijective proof to prove that |S| is equal to the expression you had in part (b),
(c)
where
S (α, β) : αε ,β ε Β, |α | + |β|- n ,
where Ja| denotes the length of a.
(note. It is not a coincidence that what you get in part (b) gives you the same answer for the question
in part (c) which you solved using elementary counting. In week 3, you will learn how to approach such
counting problems, and more complicated counting problems, with the machinery of generating series.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a130b5e-d7ab-4a36-b51e-cf5b39b169cd%2F564e1148-ca50-4f9a-bd1c-febaca7cd745%2Ft49uver_processed.png&w=3840&q=75)
Transcribed Image Text:Question 3.
of binary strings containing exactly three 1s. Let w be the weight function that returns the length of a
binary string.
Let A be the set of binary strings containing exactly two 1s. Let B be the set
(a)
Write down the generating series PA(x) and PB(x) for A and B respectively with respect
to weight w, in closed form (i.e. without using E or +·+).
(b)
Let +(x) = PA(x)ÞB(x). Express [x"]¤(x) as a function of n, for every nonnegative integer
п.
Using a bijective proof to prove that |S| is equal to the expression you had in part (b),
(c)
where
S (α, β) : αε ,β ε Β, |α | + |β|- n ,
where Ja| denotes the length of a.
(note. It is not a coincidence that what you get in part (b) gives you the same answer for the question
in part (c) which you solved using elementary counting. In week 3, you will learn how to approach such
counting problems, and more complicated counting problems, with the machinery of generating series.)
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