COMP9020 Assignment 2 2022 Term 3
Objectives and Outcomes
The purpose of this assignment is to build your mathematical maturity in the areas of Equivalence Relations,
Functions, Graph Theory, Recursion and Induction. 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 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 2.
Problem 2: 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 3: 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 3,
where we will model it with Propositional Logic.
Problem 4: The aim of this question is to show how you can prove and disprove whether or not a graph has
a particular graph property (in this case, planarity). While the specific technique of finding/excluding
subdivisions is not necessarily applicable to other graph properties, the process illustrates the general
principle of how you might prove a “negative” property.
Problem 5: The similarity between relational composition (;) and transitivity was remarked upon in lectures.
This problem clarifies that connection, showing how relational composition can be used to recursively
“add” transitivity to a non-transitive relation. This question ties together several topics: Relations,
Recursion, Induction (and, obliquely, Formal Languages), as well as results from the first Assignment;
and reinforces the basic concepts of those topics.
Problem 6: The binary tree data structure is commonly taught in introductory Computer Science and should
be familiar to most students taking COMP9020 (though it is not essential to know about them for the
purposes of this question). In this question we take an abstract model of a binary tree and use this
to create abstract, recursive “programs” (recursively defined functions) and then reason about these
programs. We will revisit this question in Assignment 3 when we use binary trees themselves as
abstract models (of logical formulas) and utilise the recursive definition to count these objects.
Problem 7: In Problems 5 and 6, the necessary recursive definition has been provided. The aim of this ques-
tion is to create your own recursive definition, providing a formal definition of the previously informally
defined lexicographic order. This provides a connection between recursion and partial orders, further
highlighting the ubiquity of recursion in many parts of Computer Science.
After completing this assignment, you will:
Be able to make logical arguments about abstract concepts [Problems 1, 2, 4, 5 and 7]
Be able to create and reason about abstract models of concrete (“real-world”) problems and objects
[Problems 3, 6 and 7]
Understand several deeper connections between different topics of the course [Problems 1, 2, 5 and 7]
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Due: Monday, 31st October, 12:00 (AEDST)
Submission is through WebCMS/give and should be a single pdf file, maximum size 2Mb. Prose
should be typed, not handwritten. Use of LATEX is encouraged, but not required.
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 (12 marks)
Let S be a set.
(a) Show that for any set T and any function f : S! T, the relation Rf ? S? S, defined as:
(s, s0) 2 Rf if and only if f (s) = f (s0)
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, s0) 2 R if and only if fR(s) = fR(s0)
6 marks
Problem 2 (20 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 2 N:
(i) f (a+ b) = max{ f (a), f (b)} 3 marks
(ii) f (ab) = min{ f (a), f (b)} 3 marks
From Problem 1, we know that Rf ? N?N, the relation given by:
(m, n) 2 Rf if and only if f (m) = f (n)
is an equivalence relation. Let E ? Pow(N) be the set of equivalence classes of Rf , and for n 2 N, let
[n] 2 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].
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The difficulty is that the operands [x] and [y] can have multiple representations (e.g. if z 2 [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) 2 if and only if there is x, y 2 N such that X = [x], Y = [y] and Z = [x+ y]
((X,Y),Z) 2 if and only if there is x, y 2 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 2 E:
(i) A [1] = A 3 marks
(ii) A B = B A 3 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 3 (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
Problem 4 (12 marks)
This is the Petersen graph:
(a) Give an argument to show that the Petersen graph does not contain a subdivision of K5. 6 marks
(b) Show that the Petersen graph contains a subdivision of K3,3. 6 marks
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Problem 5 (20 marks)
Let R ? S? S be any binary relation on a set S. Consider the sequence of relations R0,R1,R2, . . ., defined
as follows:
R0 := I = {(x, x) : x 2 S}, and
Rn+1 := Rn [ (R;Rn) for n 0
(a) Prove that for all i, j 2 N, if i ? j then Ri ? Rj. Hint: Let Pi(j) be the proposition that Ri ? Rj and prove
that Pi(j) holds for all j i. 4 marks
(b) Let P(n) be the proposition that for all m 2 N: Rn;Rm = Rn+m. Prove that P(n) holds for all n 2 N.
Hint: Use results from Assignment 1 4 marks
(c) Prove that if there exists i 2 N such that Ri = Ri+1, then Rj = Ri for all j i. 4 marks
(d) If |S| = k, explain why Rk2 = Rk2+1. 2 marks
(e) If |S| = k, show that Rk2 is transitive. 2 marks
(f) If |S| = k show that Rk2 is the minimum (with respect to ?) of all reflexive and transitive relations that
contain R. 4 marks
Remark
The relation at the limita as n tends to infinity, R? = limn!? Ri, is known as the reflexive, transitive
closure of R, and is closely connected to the Kleene star operator.
aBecause Rj ? Ri ? S? S for all j ? i, the Knaster-Tarski theorem ensures this limit always exists, even for infinite S.
Problem 6 (20 marks)
A binary tree is a data structure where each node is linked to at most two successor nodes:
If we include empty binary trees (trees with no nodes) as part of the definition, then we can simplify the
description of the data structure. Rather than saying a node has 0, 1, or 2 successor nodes, we can instead
say that a node has exactly two children, where a child is a binary tree. That is, we can abstractly define
the structure of a binary tree as follows:
(B): An empty tree, t
(R): An ordered pair (Tleft, Tright) where Tleft and Tright are trees.
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So, for example, the above tree would be defined as the tree T where:
T = (T1, T2), where
T1 = (T3, T4) and T2 = (T5, t), where
T3 = T4 = T5 = (t, t)
That is,
T =(t, t), (t, t)
(t, t), t
A leaf in a binary tree is a node that has no successors (i.e. it is of the form (t, t)). A fully-internal node in
a binary tree is a node that has exactly two successors (i.e. it is of the form (T1, T2) where T1, T2 6= t). The
example above has 3 leaves (T3, T4, and T5) and 2 fully-internal nodes (T and T1). For technical reasons
(that will become apparent) we assume that an empty tree has 0 leaves and 1 fully-internal nodes.
(a) Based on the recursive definition above, recursively define a function count(T) that counts the number
of nodes in a binary tree T. 4 marks
(b) Based on the recursive definition above, recursively define a function leaves(T) that counts the number
of leaves in a binary tree T 4 marks
(c) Based on the recursive definition above, recursively define a function internal(T) that counts the num-
ber of fully-internal nodes in a binary tree T. 4 marks
(d) If T is a binary tree, let P(T) be the proposition that leaves(T) = internal(T)+ 1. Prove that P(T) holds
for all binary trees T. Your proof should be based on your answers given in (b) and (c). 8 marks
Problem 7 (4 marks)
Let S be a finite set, totally ordered by <. Give a formal, recursive definition of the lexicographic ordering