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COMP2121A: Discrete Mathematics
创建日期：2024-05-15 17:22:30
标签： COMP2121A
Discrete Mathematics

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COMP2121A: Discrete Mathematics

Rules: Discussion of the problems is permitted, but writing the assignment together is not
(i.e. you are not allowed to see the actual pages of another student).
Course Outcomes
• [O1. Abstract Concepts]
• [O2. Proof Techniques]
• [O3. Basic Analysis Techniques]
Hint: You may write programs for some counting questions.
1. (10 points) [O1, O3] Let X = {x1, x2, ..., xm} and Y = {y1, y2, ..., yn}.
(a) How many different relations are there over X and Y ? (The relation defined on
two sets is a subset of their Cartesian product. Mathematically, the Cartesian
product of set A and B denoted by A×B is equal to {(a, b)|a ∈ A and b ∈ B}.)
(b) How many different functions are there from X to Y ?
(c) What is the condition m and n should satisfy to make injective functions exist?
What is the number of injective functions under the condition (in terms of m
and n)?
(d) What is the condition m and n should satisfy to make surjective functions exist?
What is the number of surjective functions under the condition (in terms of m
and n)?
(e) What is the condition m and n should satisfy to make bijective functions exist?
What is the number of bijective functions under the condition (in terms of m
and n)?
2. (10 points) [O3] Assume we have 7 apples, 4 oranges, and 1 banana. The apples and
oranges are identical separately. Please count the number of all permutations for each
of the following cases:
(a) put these fruits in a row;
(b) put these fruits in a circle; (orientation matters, i.e., clockwise and counter-clock
are different.)
(c) put these fruits in a row where the banana cannot be adjacent to any oranges.
3. (10 points) [O3] There are 5 balls in an opaque box, whose colors are white, red, blue,
green and yellow, respectively. They are not distinguishable unless being taken out.
Each time Amy uniformly at random takes a ball out from the box (with replacement).
(a) What is the expected number of times of taking to get the first two consecutive
white balls?
1
(b) What is the expected number of times of taking to get the first consecutive white
and red balls? (a white ball immediately followed by a red ball)
4. (10 points) [O3] There are 1000 coins, among which 999 are fair, i.e., have a head
and a tail and the probability of getting a head or a tail is equal, while 1 has heads
on both sides. Amy is playing a game: in each round, Amy picks a coin uniformly
at random and then tosses it 10 times. What is the probability of the selected coin
being unfair if Amy gets 10 heads in a round?
5. (10 points) [O3] A group of 2N people stand in a line in front of a vending machine
to buy drinks, among which each of N people holds a $5 coin, and each of the other
N people holds a $10 coin. Everyone wants only one can of drink. Drinks are all sold
at a price of $5. If the vending machine cannot give change, i.e., a person puts in
a $10 coin while there are no $5 coins in the machine, it will stop working at once.
Assume at the beginning there are no coins in the machine. People holding coins with
the same denomination are considered indistinguishable.
(a) How many queuing ways are there such that the machine will stop working when
N = 3?
(b) Let A be the set of all queuing ways that will make the machine stop working.
Each element in A consists of N 5’s and N 10’s, for example, if N = 2, an
element {5, 10, 10, 5} ∈ A denotes the first and forth person holding $5 coins and
the second and third person holding $10 coins. The machine will stop working
when the third person puts in the coin in this case. Define a function f , which
maps each element a ∈ A to a sequence of length 2N as follows: if the machine
stops working when the ith person puts in the coin in the queuing way a, then in
f(a), the first i numbers would keep unchanged while the following 5’s would be
replaced by 10’s and 10’s would be replaced by 5’s. Let f(A) = {f(a)|∀a ∈ A}.
What is the cardinality of |f(A)|?
(c) How many possible queuing ways are there such that the machine would not stop
working?

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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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