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COMP 2711 Discrete Mathematical

创建日期：2024-05-09 14:50:06

所属分类：计算机computer science

标签： COMP 2711

Discrete Mathematical

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COMP 2711 Discrete Mathematical Tools for Computer Science

Note 1: For all non-proof questions, showing your steps is not necessary unless
required otherwise. However, in case your answer is wrong, showing your steps
may earn you some partial credits.
Note 2: You can express your answers using binomial coefficients, factorials, and
Dn (derangement number). However, you should not have summation
∑
in your
final answers. For example,
(
10
3
)
D9+4! and 1!+2!+3!+4! are valid, but
∑n
i=0
(
n
i
)
or
(
n
0
)
+
(
n
1
)
+ · · ·+ (n
n
)
is not. The latter has to be simplified to 2n.
Question 1: [10 pts] You are constructing a string of n digits by the following procedure.
1. Set all the n digits to 0.
2. For every x1-th digit, replace it by 1.
3. For every x2-th digit, replace it by 2.
4. For every x3-th digit, replace it by 3.
Below is an example with n = 15, x1 = 3, x2 = 5 and x3 = 7.
“001021301201032”
Suppose x1 = 31, x2 = 43 and x3 = 53. What is the minimum value n so
that the string ends with “31020”?
Solution: This is equivalent to finding the smallest n that satisfies the following system
of congruences.
n ≡ 3 (mod 31)
n ≡ 1 (mod 43)
n ≡ 4 (mod 53)
Let m = 31 · 43 · 53 = 70649,M1 = 43 · 53 = 2279,M2 = 31 · 53 = 1643 and
M3 = 31 · 43 = 1333. By the extended Euclidean algorithm, we have 2 is
an inverse of M1 (mod 31), 24 is an inverse of M2 (mod 43) and 20 is an
inverse of M3. By the Chinese Remainder Theorem, n = 3 · 2 ·M1 + 1 · 24 ·
M2 + 4 · 20 ·M3 = 159746 ≡ 18448 (mod 70649). So, the minimum length
is 18448.
Question 2: [10 pts] There are n coins, all of equal shape and size, but only n − 1 of
them are of equal weight. You are given a balance scale which can compare
the weight of any two sets of coins. Assume n is a power of 3. We use
the following algorithm to find the coin of different weight. Show that the
number of weightings is at most 2 log3 n.
Procedure findCoin(S : a set of coins)
if |S| = 1:
return the only coin in S.
Partition S equally into 3 sets of coins, A,B and C.
if A and B are of equal weight:
return findCoin(C)
else if B and C are of equal weight:
return findCoin(A)
else:
return findCoin(B).
Solution: In each recursive call, it takes at most 2 comparisons and the input size are
reduced to one-third of the parent. So, the number of comparisons is at
most 2 log3 n = O(log n).
Question 3: [10 pts] Consider a company with n > 0 workers. A worker is a famous-solo-
worker if she or he is known by every other worker, but knows none of them.
You are the boss and you want to check if there is a famous-solo-worker
in your company. You are only allowed to ask the following question to a
worker: “Do you know worker j?” The workers will answer your questions
truthfully. Show that you can find the famous-solo-worker, or declare that
such a worker doesn’t exist, with at most 3(n− 1) questions.
Note that there can be at most one famous-solo-worker.
Solution: We prove this by induction.
Basis step:
When n = 1, the only worker must be a famous-solo-worker, and no ques-
tions are needed, which equals to 3(1− 1) = 0.
Inductive step:
The inductive hypothesis is that if there are k ≥ 1 workers, then we can
determine whether there is a famous-solo-worker with at most 3(k − 1)
questions.
Suppose there are k + 1 workers. Let i and j be any two workers among
them. We ask whether i knows j, this takes one question. If i knows j, then
i is not a famous-solo-worker. If i doesn’t know j, then j is not a famous-
solo-worker. Without loss of generality, assume that i is not a famous-solo-
worker. Excluding i from the k + 1 workers, there are k workers (including
j) remaining. By the inductive hypothesis, we can use 3(k − 1) questions
for our checking.
If there is no famous-solo-worker among these k workers, then we know
that there is no famous-solo-worker, and the number of questions used is
1 + 3(k − 1) = 3k − 2 ≤ 3k.
If there is a famous-solo-worker s among these k workers, then we ask
whether s knows i and whether i knows s. This takes two more questions.
If i knows s and s doesn’t know i, then s is still a famous-solo-worker,
otherwise s is not a famous-solo-worker (thus no famous-solo-worker). In
this case, we used a total of 1 + 3(k − 1) + 2 = 3k questions.
Question 4: [8 pts] Suppose that P (n) is a predicate. Under each of the following
conditions, find the integers n such that P (n) must be true:
(a) P (0) is true; for all positive integers n, if P (n) is true, then P (n + 2)
is true.
(b) P (4) is true; for all positive integers n, if P (n) is true, then P (2n) is
true.
(c) P (1) is true; for all positive integers n, if P (n) and P (n+ 1) are true,
then P (n + 2) is true.
(d) P (2) is true; for all positive integers n, if P (k) is true for 2 ≤ k ≤ n,
then P (n + 1) is true.
Solution: (a) 0 only.
(b) For all integers n that is a power of 2 and greater than or equal to 4.
(c) 1 only.
(d) For all integers n ≥ 2.
Grading scheme: 2 points for each.
Question 5: [6 pts] Give a recursive defintion of the set of bit strings that have more 0s
than 1s.
Solution: Let S be the set of bit strings that have more 0s than 1s.
Base case: 0 ∈ S.
Recursive case: If x and y are bit strings in S, then xy ∈ S, xy1 ∈ S,
x1y ∈ S and 1xy ∈ S.
Question 6: [10 pts] A string is called E-String if the string consists of only digits in
{0, 2, 4, 7, 8, 9} and the sum of the digits is even. E.g. “079” is an E-String
because 0+7+9 = 16 is even. “708” is not an E-String because 7+0+8 = 15
is odd. “107” is not an E-String because the digit 1 is not allowed. The
empty string is also considered an E-string.
How many E-Strings of length n are there? Give a closed-form solution.
[Hint: use a recurrence.]
Solution: Let E(n) denote the number of E-Strings of length n.
Base case: E(0) = 1
Recursive case: For n > 0, E(n) = 2 · (6n−1 −E(n− 1)) + 4 ·E(n− 1) =
2 · 6n−1 + 2 · E(n− 1).
E(n) = 2 · 6n−1 + 2 · E(n− 1)
= 2 · 6n−1 + 2 · (2 · 6n−2 + 2 · E(n− 2))
= 2 · 6n−1 + 22 · 6n−2 + 22 · E(n− 2)
...
=
n∑
i=1
2i · 6n−i + 2n · E(0)
=
n∑
i=1
2n · 3n−i + 2n
= 2n
n−1∑
i=0
3i + 2n
= 2n
3n − 1
3− 1 + 2
n
= 2n−1(3n − 1) + 2n

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 101H1 IB9520 FINM 3008 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