# Let s be the following element of BT : Draw the trees (i) MAKE LEFT s (∅,10, ( )) (ii) MAKE RIGHT LEFT s ROOT LEFT s LEFT LEFT s ( ( ( ) , ( ) , ( ) ) ( ) ( ))

## Discrete Mathematics

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Question 1 (5 marks)
Relations R and S on the set {a b c d , , , } are represented respectively by the
matrices
F T T F
F T T F
F F F T
T F F F
a b c d
a
b
c
d
 
 
 
 
   

T F F F
T T F F
F T F T
F F T T
a b c d
a
b
c
d
 
 
 
 
   
(a) List the ordered pairs belonging to R .
(b) Determine the matrix representing the composition S R .
Question 2 (5 marks)
Let f and g be functions from ℕ to ℕ, where ℕ is the set of natural numbers,
with
( ) 1
( ) 2
f n n
g n n
= +
=
Determine f f  , f g and g f  .
8606 ASSIGNMENT 2 PAGE 2
Question 3 (5 marks)
(a) A palindrome is a string of digits which reads the same backwards as forwards.
How many different palindromes are there with 9 digits?
(b) How many distinct rearrangements are there of the letters in the word
MATHEMATICS
(i) begin with the letter H
(ii) have both the M together?
Question 4 (5 marks)
(a) Use the minimal spanning tree algorithm to find a minimum connector for the
graph G below.
A
4 7
5 6
B 5 C
4 3 2 8

D
5 7
E F
6
(b) Explain how you would amend the algorithm to find a spanning tree of maximum
total weight. Hence, find a spanning tree of maximum total weight for the graph G
above.
8606 ASSIGNMENT 2 PAGE 3
Question 5 (5 marks)
(a) Use Dijkstra’s Algorithm to find the shortest distance from vertex A to all the
other vertices in the following weighted digraph:
D
4 4 21
B F
6
13
A 5 3 H
5
11
C G
22 5 4
E
(b) Determine two different shortest paths from A to H .
8606 ASSIGNMENT 2 PAGE 4
Question 6 (10 marks)
The set BT consists of all binary rooted trees whose vertices are elements of the
set ℕ of natural numbers. The empty tree is denoted by ∅ .
The following basic primitive functions for manipulating elements of BT are given:
ROOT BT : → ℕ, where ROOT t( ) = the root of the non-empty tree t
LEFT BT BT : → , where LEFT t( ) = the left subtree of the non-empty tree t
RIGHT BT BT : → , where RIGHT t( ) = the right subtree of the non-empty tree t
MAKE BT BT BT : × × → ℕ , where MAKE t n s ( , , ) = the tree with root n , left
subtree t and right subtree s
(a) Let s be the following element of BT :
Draw the trees
(i) MAKE LEFT s (∅,10, ( ))
(ii) MAKE RIGHT LEFT s ROOT LEFT s LEFT LEFT s ( ( ( ) , ( ) , ( ) ) ( ) ( ))
(b) A function G BT : → ℕ is defined as below.
G t( ) 0 = if LEFT t RIGHT t ( ) ( ) = = ∅
G t G RIGHT t ( ) = +1 ( ( )) if LEFT t( ) = ∅ and RIGHT t( ) ≠ ∅
G t G LEFT t ( ) = +1 ( ( )) if LEFT t( ) ≠ ∅ and RIGHT t( ) = ∅
G t G LEFT t G RIGHT t ( ) = + + 2 ( ( )) ( ( )) otherwise
(i) Evaluate G s( ) where s is the tree in part (a).
(ii) Describe the general effect of G on any tree in BT .
(c) The function SWAP BT BT : → interchanges the root of an input tree t and
the root of its left subtree (provided LEFT t( ) ≠ ∅ ). Give a description, in
terms of the basic primitive functions above, of the function SWAP .
9
5
2
4 3 7
8 1
8606 ASSIGNMENT 2 PAGE 5
Question 7 (10 marks)
The following algorithm, Algorithm1, counts the number of pairs of integers in a list.
Algorithm Algorithm1
begin
Input x1, x2, …, xn
count := 0
for i := 2 to n do
begin
for j := 1 to (i – 1) do
begin
if xi = xj then
begin
count := count + 1
end
end
end
Output count
end
(a) Verify that the algorithm works on the list 2, 3, 6, 2, 6.
(b) Find a time complexity function for the algorithm by calculating how many times
the comparison i j x x = is performed for an input sequence of length n .
(c) A second algorithm, Algorithm2, also counts the number of pairs of integers in
a list. If Algorithm2 is uses 2
n n log operations, which of the two algorithms is
more efficient? Briefly justify your answe

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## A++ SOLUTION TO ALL QUESTIONS

Course

Name

Institution Affiliation

1. a)

Orders pairs in R

b) SoR

a b c d a F T T F b F T T F c F F F T d T 0 0 F a b c d a T F F F b T T F F c F T F T d F