# Mathematics M Assignment 2012

*Mathematics for Computer Scientists* teaches you the basic logical and mathematical theory that is necessary to become a good programmer and computer scientist. It covers some familiar fields, like Arithmetic and Algebra, but also parts of Math that are more specific to the study of data structures and algorithms. First of all, it is necessary to learn the fundamental reasoning techniques: how to prove mathematical statements and how to check that a proof is correct. Then you must become familiar with the concepts of set, function, relation and be able to work with them with confidence. All this forms the basis for specific mathematical theories. Arithmetic is the first one: the most important method of proof here is induction on natural numbers. Then comes Boolean Algebra, essential to understand the logic of circuits and data structures. Other topics of study are Combinatorics, which studies the way to count complex arrangements of objects, and Modular Arithmetic, the "Mathematics of clocks". At the end of the module you will have acquired the intellectual tools that allow you to understand the abstract nature and the workings of algorithms.

## Outline of lectures

In this section you will find, after each lecture, a list of topics that were taught and links to additional material and lecture notes. I'm constantly trying to improve these notes, so I encourage you to tell me if you find mistakes or things that are not clearly explained.Date | Topics | Lecture Notes |
---|---|---|

1. 3 Oct 2012 | Introduction to the Module Some mathematical puzzles | Mathematical Puzzles |

2. 4 Oct 2012 | Propositions and Derivations | Propositional Logic |

3. 10 Oct 2012 | Propositional Formulas Rules for Connectives | ... |

4. 11 Oct 2012 | All Deduction Rules | ... |

5. 17 Oct 2012 | Truth Tables Logical Equivalence | Boolean Algebra |

6. 18 Oct 2012 | Tautologies, Boolean Algebra | ... |

7. 24 Oct 2012 | Derivation of Boolean Equalities Numbers Systems, Order | Arithmetic |

8. 25 Oct 2012 | Partial Order, Divisibility Universal and Existential Quantifiers | ... |

9. 31 Oct 2012 | Floor, Ceiling, Absolute Value Arithmetic: Recursive Functions | ... |

10. 1 Nov 2012 | Fibonacci Function, Greatest Commond Divisor | ... |

11. 7 Nov 2012 | Summations | ... |

12. 8 Nov 2012 | Induction | ... |

13. 14 Nov 2012 | Sets, Venn Diagrams Cartesian Product, Subsets | Sets, Functions, Relations |

14. 15 Nov 2012 | Cardinality, Inclusion-Exclusion Principle | ... |

15. 21 Nov 2012 | Injective/Surjective/Bijective Functions Inverse Functions | ... |

16. 22 Nov 2012 | Combinatorics: The pigeonhole principle Counting Functions and Subsets | Combinatorics |

17. 28 Nov 2012 | Binomial Coefficients | ... |

18. 29 Nov 2012 | Pascal's Triangle Modular Arithmetic | Modular Arithmetic |

19. 5 Dec 2012 | Induction with two base cases The Monkey and the Coconuts | ... |

20. 6 Dec 2012 | Exam Preparation | Model Exam |

End of lectures | ||

13 Dec 2012 | 10:00 - 17:00 : Office hours | ... |

2009-2010 Exam, 2009-2010 Resit Exam, 2010-2011 Exam,

**2011-2012 Exam and Solutions.**

## How to study

Here are my suggestions on how you should study and prepare for the exam. There are three activities that will help you get the best results:

- Read and study the lecture notes;
- Try to solve the coursework assignments;
- Try to solve the past exam papers.

**lecture notes**. When you try to solve the

**coursework assignments**, do it from scratch, without looking at the solutions and feedback that you received the first time that you did them. You should reach a level where you can do them without support. If this doesn't happen yet, you can go back and read your notes and study the part of the lecture notes that deal with the topic of the assignment. Then, on a different day, try it again. Repeat this process until you can do the exercises confidently.

Use a similar strategy when doing the **past exams**. You should give yourself a couple of hours to solve a complete exam paper. Sit at your desk with it and without books or notes and try to do it. Afterwards check your answers: the complete solution to last year's exam is given above.

## Suggested reading

The main study material for the module are the lecture notes that you find on this page. If you think you need more explanations and examples or if you want to learn more advanced topics, here are some very good books:- Roland Backhouse,
*Algorithmic Problem Solving* - Steven G. Krantz,
*Discrete Mathematics Demystified* - Rowan Garnier and John Taylor,
*Discrete Mathematics* - Norman L. Biggs,
*Discrete Mathematics*

*twin*of MCS: It gives more motivation and examples of application of the theory. It uses games and puzzles to teach the material. The second book is a simple introduction to all the basic mathematical notions. It doesn't require much previous background and is written in a plain easy style. The last two books present the topics in more depth and give many useful examples; they also cover more advanced material that is not treated in this module.

These texts, however, are not a replacement for the lecture notes. If you don't understand something and you find that the lecture notes are not clear enough, the right thing to do is to come to me and I will try to explain things better and expand the lecture notes.

## Coursework and Tutorials

Every two weeks on Thursday, you will find an assignment on this page. You have to solve the given problems and hand you solutions in by the following Wednesday at 16:00 at the school office (stamp the assignment and put it in the *letter box*).

The following week the teaching assistants will demonstrate the solutions during the tutorials. You have been divided into five groups, each having a tutorial on a different day and time. Check this timetable to see when and where your tutorial session is. If you're unable to attend the tutorial you've been assigned to, contact one of the teaching assistants, Laurence Day or Bas van Gijzel and ask to change your group.

Write on the cover letter of the assignment what group you belong to.

The average mark of the four assignments will be your *coursework mark*, which counts for 25% of you final mark.

** The first tutorial takes place in the week 8-12 October: It will give you a chance to ask any question about the first few lectures. **

** Be sure to read carefully the regulations about submission of coursework and plagiarism in the student handbook. In addition to those regulations, assignments must be submitted at the latest by the end of the week of their official deadline and standard penalties apply. **

## Contacting the teacher

Please don't hesitate to ask me any question you may have about the module. You can contact me by e-mail: put*G51MCS*in the subject line, so I know immediately that it is about this module.

Joscphine kit-{gm Assignment 1A Total Marks = 70 Please show all your work. Some questions require you to use graphs. tree diagrams, formulas, and so on that are difficult or impossible to reproduce in word-processing programs, so you will have to print the entire assignment and fill it out and mail it to your tutor. As an alternative. to mailing, you might be able to photograph or scan your assignment and submit it through the appropriate drop box. Ifyou do, make sure the material you are submitting is legible. Question 1 A random sample of 20 days showed the following number of cardiograms done each day at an outpatient testing centre: 45 25 39 5 42 48 35 12 4016 . 36 27 37 30 18 22 32 54 44 29 8 marks Construct a frequency distribution for these data using a lower limit for the ﬁrst class of 5 and a class width of 10. Indicate the class limits, boundaries, midpoints, frequencies and cumulative frequencies. Note: As this data set does not contain fractional values, do not use the “less than” method described on page 43 of the textbook for writing the classes. . ‘ Olga lei s \ {Weiss “Aidpolh‘l (Xm’) Mldcom‘l‘s _ lower lim‘fl 0F W's“ class '5 S x?“ : lMPYﬂNS “mil Still :% a $5 class void-Ho g l0 Wiley- class limiT ISiQ‘l: :‘l ._. [$5 2 1% Z Ilsa-t: 3.2361,; @Lﬁh‘a, ‘i.|0,l|,@13,1|i :H '55t4‘l': rig/“39,5; l5® H.19,90.3I@. ”Ba-”t :m ‘lSHH = 9?]; : cm; @2 ' @3‘ a w @333” = M ($325? hail) TAllij F Olauﬁoundarﬁ cgixthclﬁidpoin'i ecimgdaﬁ‘bt @aﬁtseeﬁmate‘y tut-H 5‘“? u 2 ”In” lo M 2 ‘B'M'HQE qq,§o,9,5),53 - "l I “1.521%.5 to M5 3+3 «5' IE will} it 3 - ‘ .26: g '0 25154 mi 5 Aims -3'I.S 0 3(1-S fife-$141- 3§~4tl nit-ii 4 314.5: -uq.s l0 4". 5 +31” — 41:11} I" 3 INS-Sling l0 . n+3 443-..10 —-——___.____ Introduction to Statistics . [J’WﬂIosephl he 1/0 “.‘J ‘h+ . Question 2 The amount a mother spends on prenatal care can have a longvterm impaet in terms of redueing eomplieations resulting from a baby-"5; low hirlh weight. A random sample of 50 new mothers was asked to estimate how much they spent on prenatal care. The results. are presented in the frequency distribution below. Amount Spent on Mothers Prenatal Care (8) (Number) 0—99 2 100— 199 4 200—299 8 300-399 20 400—499 to Total 50 3 marks 3. Calculate the mean amount spent on prenatal rare. 5 marks b. Use the short-cut formula to calculate the standard deviation of . amounts spent on prenatal eare. -2. m "‘9 V" l: (3- Mean —; 2 m JP ___:F_ 3 arm - 61‘! = 4‘15 3(qq. s)— cm (3‘11: 4‘100-5 n ,_ ' ‘ laws 4 I00+I°i¢i-1;m—.Lm; 40%;) say my"- 2940: - gpqqg) m1: I‘m".- ‘l‘isocz. W“. 3 ic0+aqq 435 :4!“ 5 3;: 50 20 300+ 3"”‘33 3% 5" 20(34‘1 5): MW 6910:: 2:35;: l ' ,l ’ - 1: at if; “00* ‘l‘t‘tzgq‘f , W‘LS lb (4% 5‘): :H‘ll 5° ‘2'- , GMVHZS b. S =‘ltm1‘c ’(émpgl _,___._.D_ “‘l 8:4 ”[930],“ t \l («.2 (93:111.: _ (1193401 .________s_°_ m 50—: = “6209111.: -ﬁawsns o ‘ "" :ll $12900 “HI—- Mathematics 215 /Assa'gmuent1A II .‘J‘ I P’MA‘— (-24 4 '4Ease-Phi ac Ulric) h-l' fret uenc rat each mint 0n the mlvrnn. - - l . ; l; I l l l l l l 4 guest... .....JJ‘I. i l J; . . 1; 0.11037, ﬁlm} J .1 0n}- 31 0:er . , _ . : E mack, L i 9132:- . 049$ ; l ; ' g 0.107. --l ....l. l ‘ l l l 3 0'15 4; Olib l l - l +-» l l l l 017‘ .’.,1 ""v'""'.' .. .l. ... . ’l L°fit_l1l e l l i i lOtUPV I lﬂ35 ”Imff I " l 1 .A Find alas: rntclfcwds ()(m J ; + J .r (in; . L r 4 marks c. Construct a relative freguency polvgon ()fnmounts spent on prmmtaﬂiﬂ care. Use ruler and the graph )elow to construct your answer. 2 Note: In addition to labeling the axes correctly, intlimte the relative. - leH ya {3 rec' ulhcr— 5.1/3.0:ch . ) _ , 4/5‘0: o-Cy zc/SOL (“'7‘ 3/50“)”; n/aor’ 03¢ . ._. .‘ , :0- . - ...... , .,,, A, ,,‘.,,, l . , l _ _ s a; if, 3:; 33:39.. 30:, ( amounf' CPCﬂ'l‘ on pmmfm/ can: ) 4 marks d. on struet a percentage ogive cantokhnt s ruler and the graph belch-to gons'trﬁct‘fxeo p nt on prenatal care. Use 1%W 3"?“ Note: In addition to labeling the axes correctly, indicate the eumulative '? MM! .9 tuna-Hit? ul _c 3‘ 2‘ 3/51: e O. M ‘t 1+4 = 6 6/50 a o .31 8 any: Ill H159 t 0- 5W 9-0 walnuts-:34 3W“: 0-“ l9 anemone: so 50,59; [.00 percentage at each point on the ogive. l l l: l L l l l l l .l l ll i l l tumulah‘re Hal. {1 V- 0.04 New. -: 11% oazwoov. c It“!- olggwtwoy. : 237a O.b$¥l°°7o 7- 687‘ l-OOXIDD‘Ie =l00'lo -—.__. cannula-{we rtLaJ—tve pwquemu.‘ : Wm- F "r'rvtal n 9+2 ‘1?“ . r t 1‘55 bound-0111 —o.5 -qq.§ ‘1‘}.5 - “MS was — 2%.5 ZQQIS— ' 3'11-5- arts—44m! Introduction to Statisticsswcrh‘m ”‘43M 2 marks e. Does the frequency polygon suggest skewness in the data? If so, in u-lnchdu-ecuon? tgj'f’s, It Pol-330.“ 5|“? We, Cl +0 HM L: 9“ 1 mark f. What percentage of mothers spent less than $200 on prenatal care? lac/0 of mohncnr, Spark [€53 {'Wah $3430 0?“ Framed-nI Cara. Mathematics 215 / Assignment 1ASosafDl’xl I‘LD. Nn'gkf Question 3 Random samples of households were, selected from each of three regions in a large metropolitan area. The number of households selected from eaeh region and the em‘responding mean household incomes are provided in the table below. Region Households Selected Mean Household Income (Number) (81,005)) ,D __ __ X , A 75 49 B 30 80 C 20 4o 4 marks What is the. mean household income for all three regions combined? ‘--——._. n “.45 x =44 Corﬁbinfd hula“; n Q to i + _ n,=Jo $1,, go (x) l l L l n3)(3 n3=ao 9‘)ch “yin-£4443 mean household income X 3'- 35“”)? 30(90)+20(2aa) ‘60? Cl“ 3 HQtDnS ”Whirled +§+ 3C+2C 35' SSS. oo ? 3675 + 24:30 + sco '“"‘“‘~ 125 = (9345 ’ IZS‘ X - 65 Question 4 4 marks A course has two quizzes of equal weight, a midterm examination that has twice the weight of a quiz, and a ﬁnal examination that has twice the weight ofthe midterm exam. Ifa student obtains 54% and 60% on the quizzes; 55% on the midterm; and 80% on the final, what will be the student's weighted mean final grade? to! _ l x , 3'9 ,.. ‘- ’ ‘- Xzzxm 32-2! maceo (04:9 ”93‘3U :- $‘l(i)+eo(r)+ 55(4)+920(4) )+l + 2 + it = 5W ‘e’ t. 987» I Introduction to Statistics

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