COMP9020代写、Python，Java程序代写

日期：2023-07-14 07:19

COMP9020 Assignment 1 2023 Term 2

Objectives and Outcomes

The purpose of this assignment is to build your mathematical maturity in the areas of Number Theory, Sets,

Equivalence Relations, Functions, and Graph Theory. Most questions are presented at a highly abstract level

so that the consequences are very general, and can be applied in a variety of situations: not just later on in the

course, but also beyond. The specific motivation for each problem can be summarised as follows:

Problem 1: This question works through a proof of Bézout’s Identity: the fact that the gcd of x and y can be

written as an integer linear combination of x and y. To do this we work through an approach common

throughout the course: Introducing a new definition based on known concepts; Explore that definition

with some examples; Prove results about the definition.

Problem 2: This question shows how we can use Bézout’s Identity (from Q1) to find multiplicative inverses

in modular arithmetic – that is, how to “divide” when working modulo y. In order to “divide” by x

(when working modulo y), we can instead multiply by a number w where wx =(y) 1. We saw this

with Quiz 1, T6: because 17× 11 =(186) 1, we can solve 17y =(186) 5 by multiplying both sides by 11:

55 =(186) 11(17y) =(186) y. We also saw a similar idea in the lectures about matrices.

Problem 3: We will revisit this result again when looking at algorithmic analysis. It is particularly useful for

analyzing the Faster Euclidean Algorithm covered in Week 2.

Problem 4: This question asks you to prove or disprove set equalities – specifically equalities involving

languages (so the laws of set operations do not apply). This question demonstrates another common

approach used in this course: requiring you to first decide whether the result is correct or not (assessing

understanding) and then requiring you to justify your stance (assessing ability).

Problem 5: This question demonstrates a close connection between several different topics already covered

in the course: functions, sets, and languages. This connection is more than coincidental – we will revisit

it again later in the course when we cover Boolean Algebras.

Problem 6: The use of exponential notation for describing sets of functions extends beyond comparing car-

dinalities. This question demonstrates the set-theoretical analogue of the rule abc = (ab)c. This has

applications in computer science as it demonstrates how functions of two inputs can be constructed

from functions of just a single input. This process is known as Currying.

Problem 7: This question shows another, very useful, characterization of equivalence relations, connecting

them with functions. This goes along with the characterization in terms of partitions (i.e. a connection

with sets) described in lectures. Most of the examples of equivalence relations given in lectures and

quizzes (e.g. Quiz 4, M1) fit this characterization; and we use the result again in Problem 8.

Problem 8: This question shows that, in certain circumstances, it is possible to “extend” addition and multi-

plication to equivalence classes. A variation of this result (for a different equivalence relation) establishes

the facts about how addtion and multiplication interact with =n stated in lectures. This particular re-

sult shows a connection between the matrix representation of relations and the adjacency matrix of their

graphical representation (the former corresponding to a matrix of equivalence classes). We will see the

semiring E again when we cover Boolean Logic.

Problem 9: The aim of this question is to model a concrete (“real-world”) problem abstractly using Graph

Theory. An abstract model lets us: identify the important aspects of a problem and ignore the extra-

neous; identify connections with other, seemingly unrelated problems; and make use of other, more

general, tools and results to solve the specific problem. We will revisit this problem in Assignment 2,

where we will model it with Propositional Logic.

After completing this assignment, you will:

Be able to make rigorous arguments about the foundational structures used in discrete mathematics

[All problems]

Be able to create and reason about abstract models of concrete (“real-world”) problems and objects

[Problem 9]

Understand several deeper connections between different topics of the course [Problems 1, 3, 5, 7, 8]

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Due: Friday, 14th July, 12:00 (AEDST)

Submission is through inspera. Prose should be typed, not handwritten.

Discussion of assignment material with others is permitted, but the work submitted must be your

own in line with the University’s plagiarism policy.

Problem 1 (16 marks)

For x, y ∈ Z we define the set:

Sx,y = {mx + ny : m, n ∈ Z}.

(a) Give four elements of S6,9. 2 marks

(b) Give four elements of S10,?16. 2 marks

For the following questions, let d = gcd(x, y) and z be the smallest positive number in Sx,y, or 0 if there

are no positive numbers in Sx,y.

(c) (i) Show that Sx,y ? {n : n ∈ Z and d|n}. 4 marks

(ii) Show that d ≤ z. 2 marks

(d) (i) Show that z|x and z|y (Hint: consider (x % z) and (y % z)). 4 marks

(ii) Show that z ≤ d. 2 marks

Remark

The result that there exists m, n ∈ Z such that mx + ny = gcd(x, y) is known as Bézout’s Identity.

Problem 2 (12 marks)

For all x, y ∈ Z with y > 1:

(a) Prove that if gcd(x, y) = 1 then there is at least one w ∈ [0, y) ∩N such that wx =(y) 1.

(Hint: Use Bézout’s identity) 4 marks

(b) Prove that if gcd(x, y) = 1 and y|kx then y|k. 4 marks

(c) Prove that if gcd(x, y) = 1 then there is at most one w ∈ [0, y) ∩N such that wx =(y) 1. 4 marks

Problem 3 (4 marks)

Prove that for all m, n ∈N>0 with n ≤ m:

3

2

(

n + (m % n)

)

< m + n. 4 marks

2

Problem 4 (12 marks)

Let Σ = {0, 1}. For each of the following, prove that the result holds for all sets X, Y, Z ? Σ?, or provide a

counterexample to disprove:

(a) (X ∪Y)? = X? ∪Y? 4 marks

(b) (X ∩Y)? = X? ∩Y? 4 marks

(c) X(Y ∪ Z) = (XY) ∪ (XZ) 4 marks

Problem 5 (12 marks)

(a) List all possible functions f : {a, b, c} → {0, 1}, that is, all elements of {0, 1}{a,b,c}. 4 marks

(b) Describe a connection between your answer for (a) and Pow({a, b, c}). 4 marks

(c) Describe a connection between your answer for (a) and {w ∈ {0, 1}? : length(w) = 3}. 4 marks

Problem 6 (4 marks)

Show that for any sets A, B, C there is a bijection between A(B×C) and (AB)C. 4 marks

Problem 7 (12 marks)

Let S be a set.

(a) Show that for any set T and any function f : S → T, the relation R f ? S× S, defined as:

(s, s′) ∈ R f if and only if f (s) = f (s′)

is an equivalence relation. 6 marks

(b) Show that if R ? S× S is an equivalence relation, then there exists a set T and a function fR : S → T

such that:

(s, s′) ∈ R if and only if fR(s) = fR(s′)

6 marks

Problem 8 (16 marks)

Let B = {0, 1} and consider the function f : N→ B given by

f (n) =

{

1 if n > 0,

0 otherwise.

(a) Show that for all a, b ∈N:

(i) f (a + b) = max{ f (a), f (b)} 2 marks

(ii) f (ab) = min{ f (a), f (b)} 2 marks

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From Problem 7, we know that R f ?N×N, the relation given by:

(m, n) ∈ R f if and only if f (m) = f (n)

is an equivalence relation. Let E ? Pow(N) be the set of equivalence classes of R f , and for n ∈ N, let

[n] ∈ E denote the equivalence class of n.

We would like to define binary operations, ? and ?, on E as follows:

[x] ? [y] := [x + y]

[x] ? [y] := [xy].

The difficulty is that the operands [x] and [y] can have multiple representations (e.g. if z ∈ [x] then

[x] = [z]), and so it is not clear that such a definition makes sense: if we take a different representation of

the operands, do we still end up with the same result? For example, suppose [1] = [2]. Then we would

want [1]? [1] = [2]? [2], but with the proposed definition above, we would have [1]? [1] = [2], and

[2]? [2] = [4], and it is by no means clear that [2] = [4].

Our next step is to show that such a definition makes sense.

(b) Define relations ?,? ? E2 ×E as follows:

((X, Y), Z) ∈ ? if and only if there is x, y ∈N such that X = [x], Y = [y] and Z = [x + y]

((X, Y), Z) ∈ ? if and only if there is x, y ∈N such that X = [x], Y = [y] and Z = [xy]

(i) Show that ? is a function. 3 marks

(ii) Show that ? is a function. 3 marks

Part (b) shows that the informal definition of ? and ? given earlier is well-defined, so from now we will

view ? and ? as binary operations on E, that is ?,? : E×E→ E.

(c) Show that for all A, B, C ∈ E:

(i) A? [1] = A 2 marks

(ii) A? B = B? A 2 marks

(iii) A? (B? C) = (A? B)? (A? C) 2 marks

Remark

Objects that have a concept of “addition” (?) and “multiplication” (?) where:

addition and multiplication are associative,

both operations have identities (see (c)(i)),

addition is commutative (see (c)(ii)), and

multiplication distributes over addition (see (c)(iii))

are known as semirings. We have already seen a number of semirings in this course:

The natural numbers with usual addition and multiplication,

Integers modulo n with addition and multiplication modulo n,

Subsets of a set X with union and intersection,

Languages with union and concatenation,

Binary relations with union and relational composition (see Assignment 1),

Matrices with matrix addition and matrix multiplication.

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Problem 9 (12 marks)

Eight houses are lined up on a street, with four on each side of the road as shown:

Each house wants to set up its own wi-fi network, but the wireless networks of neighbouring houses – that

is, houses that are either next to each other (ignoring trees) or over the road from one another (directly

opposite) – can interfere, and must therefore be on different channels. Houses that are sufficiently far

away may use the same wi-fi channel. Your goal is to find the minimum number of different channels the

neighbourhood requires.

(a) Model this as a graph problem. Remember to:

(i) Clearly define the vertices and edges of your graph. 4 marks

(ii) State the associated graph problem that you need to solve. 2 marks

(b) Give the solution to the graph problem corresponding to this scenario; and determine the minimum

number of wi-fi channels required for the neighbourhood? 2 marks

(c) How do your answers to (a) and (b) change if a house’s wireless network can also interfere with those

of the houses to the left and right of the house over the road? 4 marks

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Advice on how to do the assignment

Collaboration is encouraged, but all submitted work must be done individually without consulting some-

one else’s solutions in accordance with the University’s “Academic Dishonesty and Plagiarism” policies.

Assignments are to be submitted in inspera.

When giving answers to questions, we always would like you to prove/explain/motivate your an-

swers. You are being assessed on your understanding and ability.

Be careful with giving multiple or alternative answers. If you give multiple answers, then we will

give you marks only for your worst answer, as this indicates how well you understood the question.

Some of the questions are very easy (with the help of external resources). You may make use of

external material provided it is properly referenced1 – however, answers that depend too heavily on

external resources may not receive full marks if you have not adequately demonstrated ability/un-

derstanding.

Questions have been given an indicative difficulty level:

Pass Credit Distinction High distinction

This should be taken as a guide only. Partial marks are available in all questions, and achievable by

students of all abilities.

1Proper referencing means sufficient information for a marker to access the material. Results from the lectures or textbook can be

used without proof, but should still be referenced.