COMP9020 Assignment 2 2022 Term 2
Due: Monday, 18th July, 12:00 (AEST)
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 (15 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. (9 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 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 ∈N:
(i) f (a + b) = max{ f (a), f (b)}
(ii) f (ab) = min{ f (a), f (b)}
(6 marks)
From Problem 1, 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.
1
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? That is, if [x] = [x′] and [y] = [y′] is it the case that
[x + y] = [x′ + y′] and [xy] = [x′y′]? 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 ∈ X and y ∈ Y such that x + y ∈ Z
((X, Y), Z) ∈ if and only if there is x ∈ X and y ∈ Y such that xy ∈ Z
(i) Show that is a function.
(ii) Show that is a function.
(6 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
(ii) A B = B A
(iii) A (B C) = (A B) (A C)
(8 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.
2
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)
3
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 ∈ S}, and
Rn+1 := Rn ∪ (R; Rn) for n ≥ 0
(a) Prove that for all i, j ∈ 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 ∈ N: Rn; Rm = Rn+m. Prove that P(n) holds for all n ∈ N.
Hint: Use results from Assignment 1 (4 marks)
(c) Prove that if there exists i ∈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, τ
(R): An ordered pair (Tleft, Tright) where Tleft and Tright are trees.
4
So, for example, the above tree would be defined as the tree T where:
T = (T1, T2), where
T1 = (T3, T4) and T2 = (T5, τ),
A leaf in a binary tree is a node that has no successors (i.e. it is of the form (τ, τ)). 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= τ). 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? (5 marks)
Let Σ be a finite set, totally ordered by <. Give a formal, recursive definition of the lexicographic ordering
≤lex? Σ? × Σ?.
5
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 via WebCMS (or give) as a single pdf file.
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.
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.