LIBRARY USE
LA TROBE UNIVERSITY
SEMESTER ONE EXAMINATION PERIOD
2015
Student ID:
Seat Number:
Subject Code: MAT1DM
Paper No: 1
Subject Name: DISCRETE MATHEMATICS
Paper Name: Final examination (Section A)
lu
o
s
Reading Time: 15 minutes
Writing Time: 180 m
MAT1DM/4DM
Stream A Practice Class 4A
BOOLEAN EXPRESSIONS AND KARNAUGH MAPS
The following two expressions differ at exactly one literal: wxy
z and wx
yz. This enables
the following simplification of wxy
z + wx
y z:
(f )
(i)
(h)
wxy
z + wx
y z = wx(y + y)
MAT1DM/4DM
Stream A Practice Class 3A
Introducing Boolean Algebras
Below are the axioms of a Boolean algebra. The operation
is called complementation.
(a) a + b = b + a
+ is commutative
(b) a b = b a
is commutative
(c) (a + b) + c = a + (b + c)
+ is asso
MAT1DM/4DM
Practice Class 9A
Definition. A tree is a connected graph without cycles.
Theorem 5.1.5. A connected graph is a tree if and only if V = E + 1, where V is the
number of vertices, and E is the number of edges.
1. Which of the following graphs are
MAT1DM/4DM
Practice Class 12A
AUTOMATA AND LANGUAGES
If U and V are languages, then U V is the language cfw_uv | u U and v V .
We let U 0 = cfw_ and U = U 0 U 1 U 2 U 3 U 4 . . .
Example 1. cfw_01, 1cfw_a, bb = cfw_01a, 01bb, 1a, 1bb.
Example 2. cfw_01, 1
MAT1DM/4DM
Stream A Practice Class 5A
LOGIC AND CIRCUITS
1. Let r be John is rich and let h be John is happy. Write each of the following in symbolic
logic form using the symbols r and h and the connectives (or), (and), (not) and (implies).
(Assume poor a
MAT1DM/4DM:
Topics:
Discrete Mathematics
Lecture 6B
. sequences
. series
. summation notation
Sequences
Sequences of numbers have been studied in mathematics for centuries. Of major interest is whether or not
a sequence converges to a specific number. The
MAT1DM/4DM:
Discrete Mathematics
Lecture 7B
Topics: . growth of sequences
. dominant terms
. Big O
Growth of Sequences
Big O notation is used to describe long term growth of sequences and
functions. It can be used to describe the rate of convergence in li
MAT1DM/4DM:
Discrete Mathematics Lecture 1B
Stream B lecturer: Kevin Bicknell
Room 219 in Physical Sciences 2,
email: [email protected]
Topics: Number bases
Conversion between number bases
Bit strings
What is the simplest number base?
Base one.
Co
MAT1DM/4DM:
Discrete Mathematics
Lecture 3B
Subtraction using complements
Normalised scientific notation
Two basic arithmetic rules:
1. adding zero does not change a value.
2. multiplying by one does not change a value.
Binary Example from Lecture 2B
Bi
MAT1DM/4DM:
Discrete Mathematics
Lecture 8B
Topics: . graphs and Big O
. better c values
. Big O of two series
Big O: The formal definition.
Let f : N R and g : N R with f (n) 0 and g(n) 0 for
all n. We say f (n) O g(n) if there exist positive real number
MAT1DM/4DM:
Discrete Mathematics
Lecture 4B
Topics: . the concept of an algorithm
. writing simple algorithms in pseudocode
. track the memory of an algorithm
. polynomials in telescoping form
. Horners algorithm
Algorithms
An algorithm is a step by step
MAT1DM/4DM
Practice Class 8A
EULERIAN PATHS
An Eulerian path in a graph is a path containing each edge exactly once, and an Eulerian circuit is an Eulerian path which starts and ends at the same vertex.
Theorem 4.3.9. (i) A connected graph has an Eulerian
MAT1DM/4DM
Stream A Practice Class 2A
More Combinatorics
Recall that the multinomial coefficient r1 ,r2n,.,rk can be thought of in two ways:
(i) it counts the number of ways of putting n objects into k boxes, B1 , B2 , . . . , Bk , so that r1 go
into B1 ,
MAT1DM/4DM
Stream A Practice Class 1A
Introducing Combinatorics
In this class, unless specifically asked otherwise, you can leave answers expressed as multiplications,
powers, factorials, C(n, r), etc: for example, 263 C(15, 7).
Example. A password consis
LIBRARY USE
MAT1DM EXAM 2015 SECTION B
Student ID:
Seat Number:
Refer to the cover sheet of the Section A booklet for instructions.
Answers for Section B must be written in the spaces provided in this booklet.
Page 8B is blank for extra working if need
LIBRARY USE
MAT1DM EXAM 2013 SECTION B
Student ID:
Seat Number:
Refer to the cover sheet of the Section A booklet for instructions.
Answers for Section B must be written in the spaces provided in this booklet. If you need extra
space, use the back of th
LIBRARY USE
LA TROBE UNIVERSITY
SEMESTER ONE EXAMINATION PERIOD
2015
Student ID:
Seat Number:
Subject Code: MAT1DM
Paper No: 1
Subject Name: DISCRETE MATHEMATICS
Paper Name: Final examination (Section A)
lu
o
s
Reading Time: 15 minutes
Writing Time: 180 m
LIBRARY USE
MAT1DM EXAM 2015 SECTION B
Student ID:
Seat Number:
Refer to the cover sheet of the Section A booklet for instructions.
Answers for Section B must be written in the spaces provided in this booklet.
Page 8B is blank for extra working if need
MAT1DM/4DM
Solutions to Some Exam-type Practice Questions
1. (a) The state diagram:
start
1
S2
0
0
0, 1
S4
1
1
S1
S3
0
(b) f (S2 , 10101) = s4 , f (S3 , 0101) = S3 .
(c) cfw_0(10)n , n 0, or 0(10) .
xy
2. (a) From the Dont Care Karnaugh map
to the right,
MAT1DM/MAT4DM
ASSIGNMENT 2, 2017
A
Place your assignment solutions in the appropriate MAT1DM/4DM pigeonhole in the boxes on the third
level of the Physical Sciences 2 Building before 2pm, 30th March. The first page of your solutions must
carry (1) your na
MAT1DM/4DM
1.
Practice Class 7A
(a) Which of the graphs A, B, C, D are isomorphic to graph G? (No explanation required.)
G
A
B
a
(b) Redraw the given graph on
the second set of labeled
vertices.
C
D
b
c
f
e
a
e
c
f
b
d
Redraw
on:
d
1
1 0 2
0 0 1
2 1 0
Rec
MAT1DM/4DM
Practice Class 6A
LOGIC CIRCUITS: theory and practice.
1.
(a) Complete the I/O table on the right for the following logic circuit.
(b) Column d (the output of the circuit) should have
DNF equal to x y z + x y z + x y z + x y z. Use
the Karnaugh
MAT1DM/4DM
Practice Class 11A
All the machines in this tutorial are binary, that is, with the alphabet = cfw_0, 1. The
language of a FSM (or recognised by a FSM) is the set of words it accepts.
1. Consider the finite state machine at right. Let f be its e
MAT1DM/4DM
Practice Class 10A
The inorder traverse of a binary tree is defined recursively:
(a) traverse the left subtree of the root (if there is none, go to (b),
(b) collect the contents of the root (that is, write down the root label ),
(c) traverse th
MAT1DM/4DM:
Topics:
Discrete Maths
Lecture 2B
. Addition and subtraction in bases 2, 8 and 16.
. Multiplication in base 2.
Binary Arithmetic
For binary addition of two numbers, it is necessary to be able
to deal with adding
1 + 1;
and also
1+1+1
when a 1
MAT1DM/4DM
Discrete Mathematics
Practice Class 7B
Big O
Big O: The formal definition.
Let f : N ! R and g : N ! R with f (n) 0 and
exist positive real numbers c and M such that
f (n)
cg (n)
0 for all
g (n)
for each
n
n
2
f (n)
. We say
O g (n)
if there
M
Chapter 2: Polar form and roots of complex numbers
35
Technical aside Most polynomial equations of degree n will have exactly n complex
roots. The reason that we have to be careful here is that some equations have repeated roots.
For example we say the po