MATH1064 - maths.usyd.edu.au

Sample Exam A Semester 2 2019

The University of SydneySchool of Mathematics and Statistics

MATH1064

Discrete Mathematics for Computation

November 2019 Lecturer: Jonathan Spreer

Time Allowed: Writing - two hours; Reading - 10 minutes

Exam Conditions: This is a closed-book examination — no material permitted. Writing

is not permitted at all during reading time.

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Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seat Number: . . . . . . . . . . . . . . . . .

Please check that your examination paper is complete (13 pages) and indicate by signing below.

I have checked the examination paper and a�rm it is complete.

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This examination has two sections: Multiple Choice and Extended Answer.

The Multiple Choice Section is worth 50% of the total examination.

There are 20 questions. The questions are of equal value.

All questions may be attempted.

Answers to the Multiple Choice questions must be entered on

the Multiple Choice Answer Sheet before the end of the examination.

The Extended Answer Section is worth 50% of the total examination.

There are 5 questions. The questions are of equal value.

All questions may be attempted. Working must be shown.

University-approved calculators may be used.

THE QUESTION PAPER MUST NOT BE REMOVED FROM THEEXAMINATION ROOM.

Marker’s useonly

of 13

Sample Exam A Semester 2 2019 of 13

Multiple Choice Section

In each question, choose at most one option.Your answers must be entered on the Multiple Choice Answer Sheet.

1. Let A and B be finite sets. Given that there exists a bijective function f : A � B we

may conclude that:

(a) |A| < |B| (b) |A| ⇥ |B| (c) |A| = |B| (d) |A| ⇤ |B| (e) |A| > |B|

2. How many numbers are there between 1 and 1000 that are divisible by 15 and 2?

(a) 100 (b) 66 (c) 12

(d) 33 (e) None of the above.

3. The general solution of the recurrence relation xn = 8xn�1⌅ 16xn�2 is which one of the

following expressions, where A and B are constants:

(a) xn = A4n

(b) xn = A2n+ B8

n(c) xn = A2

n+ Bn8

n

(d) xn = A4n+ B4

n(e) xn = A4

n+ Bn4

n

4. . . . 20. (There will be 20 multiple choice questions in the real exam.)

O

O

o


Blank page for working; it will not be marked

End of Multiple Choice Section

Make sure that your answers are entered on the Multiple Choice Answer Sheet

The Extended Answer Section begins on the next page


Extended Answer Section

There are five questions in this section, some with a number of parts. Write your answersin the space provided below each part. You must show your working and give reasons foryour answers. If you need more space there are extra pages at the end of the examination

paper.

1. (a) Consider the sets A = {a, b, c, 1, 2, 3}, B = {b, c, 1, 4, 5}, and C = {1, 2, 3, 4, 5}.Write down explicitly a bijection from A \B to B ⇧C, or explain why one does not

exist.

Question 1 continues on the next page

AIB a 2,33B n C E 1 4,53

f AaB Bnc

flat 1 f127 4 f137 5


Question 1 continued

(b) Let X, Y be sets and consider functions f : X � Y and g : Y � X. Suppose that

for every y ⌃ Y we have that f(g(y)) = y. Prove that f is surjective.


Let ye Y Then taking x gey c X

we havefix f gey y

Hence f is surjective



(c) Suppose that A,B, C are sets. Prove the statement

A \ (B ⇧ C) = (A \B) ⌥ (A \ C).

(A Venn diagram is not su⇥cient, be explicit about the elements in A, B, and C.)

AVBnc ECAi A

Let c All Bnc Then X E A and

Xtc B n C

Either x to B or X E C

Either XE Al B or X E Atc

7 x e AIB v Aic

A'B7uCAic7EAiCBn

x e Ai B u Aic Hence either e AIB

or Xf Atc If xe Al B then Xtc B

and hence X E Bnc If c Aic then

EC and again X B C On the

other hand E A Altogether XE Al Bnc

All Bnc AIB v Alc


2. (a) How many arrangements are there of the letters of WOOLLOOMOOLOO? (In all

parts of this question it is not necessary to evaluate formulas involving factorials,

binomial coe⇥cients, etc.)

(b) How many arrangements are there with the three Ls together?


Ws 1 Os 8 Ls 3 Ms I1 8 3 1

arrangements 1 g z l 84

Treat L as one block Hence

If l possible arrangements



(c) How many arrangements are there with the W and M together?

Treat WM or MW as one block

Traci p I Sisia

of arrangements of arrangementscontaining MW containing Wh


3. (a) Suppose that (an)n⇥1 is a sequence defined by

a1 = 1, a2 = 3 and ak = ak�1 + ak�2 for all k ⇤ 3.

Prove that for all n ⇤ 1, we have

an ⇥�

7

4

⇥n

.


We use strong induction

Base cases a E Fa 3 E F

2461 3 1

Hyp i Assume 9 92 i ale are all true k 22

Iud step ah an Uu

aw III FtlEI tE 4

E 4116

Il If FSince the statement is true for h E 2 and

if it is true for all u Eh then it is alsotrue for ht 1 it is true for all us I



(b) We draw n mutually intersecting circles in the plane so that each one crosses each

other one exactly twice and no three have a boundary point in common. (Think of

Venn diagrams with two or three mutually intersecting sets.) We are interested in

the number of regions that the plane is divided into. For example, if n = 1, then the

plane is divided into two regions, the inside and the outside of the circle. If n = 2,

then the plane is divided into four regions.

Find a formula for the number of regions that the plane is divided into with n circles.

Prove that your formula is correct.

4 2 u 3 u 44 regions 8 regions 14 regions

claim u mutually intersecting circles createh2 u 2 regions

Proof Induction on u If u 1 then 12 1 2 2and I circle producer Z regions

Suppose u circler separate the plane into 42 4 2

regions

Add Cuthst circle Since it crosses eachof the n circles in 2 points it intersects allcircles in 2h points Each pair of cousequtivepoints splits 1 region into 2

Altogether this creates 2h more regionsU2 n 2 12in ht 1

2uti 2 and the

claim fellows by the principle of math induction


4. and 5. There will be five extended questions in the real exam.

You may use the next pages for your answers


This page may be used if you need more space for your answers

End of Extended Answer Section


Formula Page

Logic

Commutative laws: p � q q � p p ⌦ q q ⌦ p

Associative laws: (p � q) � r p � (q � r) (p ⌦ q) ⌦ r p ⌦ (q ⌦ r)

Distributive laws: p � (q ⌦ r) (p � q) ⌦ (p � r)

p ⌦ (q � r) (p ⌦ q) � (p ⌦ r)

Identity laws: p � (tautology) p

p ⌦ (contradiction) p

Universal bound laws: p ⌦ (tautology) (tautology)

p � (contradiction) (contradiction)

Negation laws: p ⌦ ¬p (tautology)

p � ¬p (contradiction)

Double negative law: ¬(¬p) p

Idempotent laws: p � p p p ⌦ p p

De Morgan’s laws: ¬(p � q) ¬p ⌦ ¬q ¬(p ⌦ q) ¬p � ¬qAbsorption laws: p ⌦ (p � q) p p � (p ⌦ q) p

Negations: ¬(tautology) is a contradiction

¬(contradiction) is a tautology

Conditional: p� q ¬p ⌦ q

Grammars: Restrictions on Productions w1 � w2

Context-sensitive (type 1): w1 = lAr and w2 = lwr, where A ⌃ N, l, r, w ⌃ (N ⌥T )�and w ↵= �; or w1 = S and w2 = � as long as S is not

on the right-hand side of another production

Context-free (type 2): w1 = A, where A is a nonterminal symbol

Regular (type 3): w1 = A and w2 = aB or w2 = a, where A ⌃ N, B ⌃ N,

and a ⌃ T ; or w1 = S and w2 = �

Properties of logarithms

logb(xy) = logb(x) + logb(y), logb(xa) = a logb(x), logc(x) =

logb(x)

logb(c)

Discrete Probability

A random variable X : S � N⌅ {0} has geometric distribution with parameter p ⌃ [0, 1] if

p(X = k) = (1⌅ p)k�1

p for each k ⌃ N⌅ {0}. In this case E(X) =1

pand V (X) =

1⌅ p

p2.

Bayes’ theorem: p(F | E) =p(F )

p(E | F ) · p(F ) + p(E | F ) · p(F )· p(E | F )

End of Examination

Sample Exam B Semester 2 2019 Multiple Choice Answer Sheet

0 0�⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

1 1�⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

2 2�⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

3 3�⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

4 4�⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

5 5�⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

6 6�⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

7 7�⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

8 8�⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

9 9�⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

Write your

SID here ⌅�

Code yourSID intothe columnsbelow eachdigit, byfilling in theappropriateoval.

Answers ⌅�a b c d e

Q1 �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅Q2 �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅Q3 �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅Q4 �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅ �⇥ ⇤⌅

The University of SydneySchool of Mathematics and

Statistics

MATH1064 Discrete

Mathematics for Computation

Family Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Seat Number: . . . . . . . . . . . . . . . . .

Attempt every question. You will not be

awarded negative marks for incorrect

answers.

Fill in exactly one oval per question.

If you make a mistake, draw a cross (X)

through any mistakenly filled in oval(s)

and then fill in your intended oval.

An answer which contains two or more

filled in (and uncrossed) ovals will be

awarded no marks.

This is the first and last page of this answer sheet

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