Description
Problem 1 (20 marks)
Recall the relation composition operator ; defined as:
R1; R2 = {(a, c) : there is a b with (a, b) ∈ R1 and (b, c) ∈ R2}
For any set S, and any binary relations R1, R2, R3 ⊆ S × S, prove or give a counterexample to disprove the
following:
(a) (R1; R2); R3 = R1;(R2; R3) (4 marks)
(b) I; R1 = R1; I = R1 where I = {(x, x) : x ∈ S} (4 marks)
(c) (R1; R2)← = R←
1
; R←
2
(4 marks)
(d) (R1 ∪ R2); R3 = (R1; R3) ∪ (R2; R3) (4 marks)
(e) R1;(R2 ∩ R3) = (R1; R2) ∩ (R1; R3) (4 marks)
Problem 2 (30 marks)
Let R ⊆ S × S be any binary relation on a set S. Consider the sequence of relations R
0
, R
1
, R
2
, . . ., defined
as follows:
R
0
:= I = {(x, x) : x ∈ S}, and
R
i+1
:= R
i ∪ (R; R
i
) for i ≥ 0
(a) Prove that if there is an i such that R
i = R
i+1
, then R
j = R
i
for all j ≥ i. (4 marks)
(b) Prove that if there is an i such that R
i = R
i+1
, then R
k ⊆ R
i
for all k ≥ 0. (4 marks)
(c) Let P(n) be the proposition that for all m ∈ N: R
n
; R
m = R
n+m. Prove that P(n) holds for all n ∈ N.
(8 marks)
(d) If |S| = k, explain why R
k = R
k+1
. (Hint: Show that if (a, b) ∈ R
k+1
then (a, b) ∈ R
i
for some i < k + 1.)
(4 marks)
(e) If |S| = k, show that R
k
is transitive. (4 marks)
(f) If |S| = k, show that (R ∪ R←)
k
is an equivalence relation. (6 marks)
1
Problem 3 (26 marks)
A binary tree is a data structure where each node is linked to at most two successor nodes:
If we allow empty binary trees (trees with no nodes), then we can simplify the description by saying a
node has exactly two children which are binary trees.
(a) Give a recursive definition of the binary tree data structure. Hint: review the recursive definition of a
Linked List (6 marks)
A leaf in a binary tree is a node that has no successors (i.e. it has two empty trees as children). A fullyinternal node in a binary tree is a node that has two successors. The example above has 3 leaves and 2
fully-internal nodes.
(b) Based on your recursive definition above, define the function count(T) that counts the number of nodes
in a binary tree T. (4 marks)
(c) Based on your recursive definition above, define the function leaves(T) that counts the number of
leaves in a binary tree T. (4 marks)
(d) Based on your recursive definition above, define the function internal(T) that counts the number of
fully-internal nodes in a binary tree T. Hint: it is acceptable to define an empty tree as having −1 fullyinternal nodes. (4
marks)
(e) If T is a binary tree, let P(T) be the proposition that leaves(T) = 1 + internal(T). Prove that P(T) holds
for all binary trees T. (8 marks)
Problem 4 (24 marks)
Four wifi networks, Alpha, Bravo, Charlie and Delta, all exist within close proximity to one another as
shown below.
Alpha Bravo Charlie Delta
Networks connected with an edge in the diagram above can interfere with each other. To avoid interference
networks can operate on one of two channels, hi and lo. Networks operating on different channels will not
interfere; and neither will networks that are not connected with an edge.
Our goal is to determine (algorithmically) whether there is an assignment of channels to networks so
that there is no interference. To do this we will transform the problem into a problem of determining if a
propositional formula can be satisfied.
2
(a) Carefully defining the propositional variables you are using, (4 marks)
write propositional formulas for each of the following requirements:
(i) ϕ1: Alpha uses channel hi or channel lo; and so does Bravo, Charlie and Delta. (4 marks)
(ii) ϕ2: Alpha does not use both channel hi and lo; and the same for Bravo, Charlie and Delta.(4 marks)
(iii) ϕ3: Alpha and Bravo do not use the same channel; and the same applies for all other pairs of
networks connected with an edge. (4 marks)
(b) (i) Show that ϕ1 ∧ ϕ2 ∧ ϕ3 is satisfiable; so the requirements can all be met. Note that it is sufficient
to give a satisfying truth assignment, you do not have to list all possible combinations. (4 marks)
(ii) Based on your answer to the previous question, which channels should each network use in order
to avoid interference? (4 marks)
