Problem Set 3 Solution

General instructions




Please read the following instructions carefully before starting the problem set. They contain important information about general problem set expectations, problem set submission instructions, and reminders of course policies.




Your problem sets are graded on both correctness and clarity of communication. Solutions which are technically correct but poorly written will not receive full marks. Please read over your solutions carefully before submitting them. Proofs should have headers and bodies in the form described in the course note.



Each problem set may be completed in groups of up to three. If you are working in a group for this problem set, please consult https://github.com/MarkUsProject/Markus/wiki/Student_Groups for a brief explanation of how to create a group on MarkUs.



Exception: Problem Sets 0 and 1 must be completed individually.




Solutions must be typeset electronically, and submitted as a PDF with the correct lename. Hand-written submissions will receive a grade of ZERO.



The required lename for this problem set is problem set3.pdf.




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The work you submit for credit must be your own; you may not refer to or copy from the work of other groups, or external sources like websites or textbooks. You may, however, refer to any text from the Course Notes (or posted lecture notes), except when explicitly asked not to.









Additional instructions




For each proof, clearly dene the predicate ( P (n)) that is relevant for your proof.



You may not use forms of induction we have not covered in lecture.



Please follow the same guidelines as Problem Set 2 for all proofs.









[12 marks] extend some results...
De nition 1 (sequence an, S). Let a : N 7!Z. Denote a(n) = an, and a is identi

ed with the sequence




a0; a1; a2; :::. Let S = ff j f : N 7!Zg be the set of integer sequences.




[3 marks] Use induction on n to prove that if m is some non-zero integer, and a; b 2 S are arbitrary integer sequences, and n is an arbitrary natural number greater than 0, then









 


k=n


k=n




8k µ n; ak bk
(mod m)
)
Y
ak
Y
bk
(mod m)










k=0


k=0







Hint: You may assume 2.18(c) from the course notes as a starting point.




[3 marks] Use induction on n to prove that if d 2 N, d 1, and b is an integer sequence with bm 0 for all natural numbers m, and n is an arbitrary natural number, then



i=n

8i 2 N; i µ n ) gcd(d; bi) = 1 ) d - Y bi

i=0




Hint: You may assume 2(g) from problem set 2 as a starting point.




(c) [3 marks] Consider the sums




1
1


14


13






1










1
1


37


13




+


=

















+




+


=







2 + 1
2 2
24
24




3 + 1
3 + 2
2 3
60
24
Use induction to prove that for all natural numbers n, if n 1 then:




























j=2n


1




13




































X























































j
24














































































j=n+1






































































(d) [3 marks] Dene integer sequence c 2 S by










































cn = 80;


1 + 3n
2
















if n = 0




















<cn






3n + 1; if n 0




















:


















3
.


















Use induction on n to prove that for all
n 2 N; cn = n





















[8 marks] Counting subsets



[3 marks]



De nition 2 (




nk!). Let




n; k




2 N,




k




µ n, and S




be a set with




jSj =




n. Then




nk!




denotes the number



of subsets S of size k.




Use induction on n to prove




8n; k 2 N; k µ n )
n
! =
n!




k
k!(n k)!



Hint: Notice that no induction is required when k = 0 or k = n, and look for a connection between

n + 1! and both
n! and
n
!. This approach requires you to introduce k after n. Anti-
k
k
k 1





Hint: You may not use results from combinatorics such as the Binomial Theorem, since they are, essentially, what you are proving.




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Problem Set 3




De nition 3
(Sn). Let n 2 N. Dene Sn = f1; 2; : : : ; ng
De nition 4
(DT Pn). Let n 2 N. Dene the set of disjoint two-set partitions of Sn as follows:


DT Pn = fA; Bg j A; B Sn and A [ B = Sn and A \ B = ;
Notice that DT P0 = f;; ;g
and DT P1 = ff1g; ;g .
[2 marks] Write out all the elements of DT P2 and DTP3 explicitly.



[3 marks] Find a closed-form expression for jDT Pnj in terms of n. Use induction on n to prove your formula correct.



[11 marks] asymptotics



[3 marks] Use the de
(b) [3 marks] Use the denition of big-Oh from the course notes to prove that for all a; b 2 R+

b a ^ a 1 ) bn 62O(an)

You may not use limits or other techniques of calculus.




(c) [5 marks] Read over function xgcd, which calculates the extended gcd(n, m), below:







def xgcd(n, m):



s1, s0, t1, t0, r1, r0 = 0, 1, 1, 0, m, n



while r1 != 0:



quotient = r0 // r1



r0, r1 = r1, r0 - quotient * r1



s0, s1 = s1, s0 - quotient * s1



t0, t1 = t1, t0 - quotient * t1



return (r0, s0, t0)






Let the input size be n. Assume that the loop body, lines 4{7, is 1 \step". Prove that the runtime of xgcd, RTxgcd 2 O(lg n). Hint: Can you show that every two iterations of the loop reduces r0 by




at least half?











































































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