SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 4
Prove each statements below by using mathematical induction.
1. Verify that each statement is true for every positive integer n.
a) 1(1!) + . + n(n!) = (n + 1)! 1
b) 1 + 3 + 5 + . + (2n 1) = n2
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 3
1. In ab, use the denitions of even, odd, prime, and composite to justify each of
your answers.
a) Assume that k is a particular integer.
i. Is 17 an odd integer?
ii. Is 0 an even integer?
iii.
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 13
1. Either draw a graph with the given specication or explain why no such graph
exists.
a) Tree, nine vertices, nine edges.
b) Graph, connected, nine vertices, nine edges.
c) Graph, circuit free
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 13
1. Either draw a graph with the given specication or explain why no such graph
exists.
a) Tree, nine vertices, nine edges.
b) Graph, connected, nine vertices, nine edges.
c) Graph, circuit free
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 12
1. Dene each graph formally by specifying its vertex set, its edge set, and a table
giving the edge-endpoint function.
a)
b)
2. A graph has vertices of degrees 0, 2, 2, 3 and 9. How many edges
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 12
1. Dene each graph formally by specifying its vertex set, its edge set, and a table
giving the edge-endpoint function.
a)
b)
2. A graph has vertices of degrees 0, 2, 2, 3 and 9. How many edges
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 11
n (n + 1)
1. The formula 1 + 2 + 3 + . + n =
is true for all integers n 1. Use this
2
fact to solve each of the following problems:
a) If k is an integer and k 2, nd a formula for the expressio
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 11
n (n + 1)
1. The formula 1 + 2 + 3 + . + n =
is true for all integers n 1. Use this
2
fact to solve each of the following problems:
a) If k is an integer and k 2, nd a formula for the expressio
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 10
1. Find the rst four terms of each of the recursively dened sequences below:
a) ak = 2ak1 + k , for all integers k 2
a1 = 1
b) ck = k (ck1 )2 , for all integers k 1
c0 = 1
c) tk = tk1 + 2tk2 ,
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 10
1. Find the rst four terms of each of the recursively dened sequences below:
a) ak = 2ak1 + k , for all integers k 2
a1 = 1
b) ck = k (ck1 )2 , for all integers k 1
c0 = 1
c) tk = tk1 + 2tk2 ,
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 9
1. Write all the 2-permutations of W, X, Y, Z .
2. Evaluate the following quantities.
a) P(6, 4)
b) P(6, 6)
c) P(6, 3)
d) P(6, 1)
3. How many ways can three of the letters of the word ALGORITHM
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 8
1. A person buying a personal computer system is oered a choice of three models
of the basic unit, two models of keyboard, and two models of printer. How many
distinct systems can be purchased?
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 7
1. For each pair of integers a and b, nd integers q and r such that a = bq + r and
0 r < |b|:
a) a = 258 and b = 12
b) a = 278 and b = 12
c) a = 313 and b = 5
d) a = 92 and b = 4
2. Find the gre
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 5
1. Let X = cfw_1, 2, 3, 4 and Y = cfw_2, 4, 5, 7, 8
a) Is 3 X ?
Is 1 Y ?
Is X Y ?
b) What is |Y |?
c) Write down set X Y .
d) Write down set X Y .
e) Write down set Y X .
2. Let the universal se
TUTORIAL
TUTORIAL 4
TUTORIAL
TUTORIAL 4
Prove the following statements by using the method of proof
by induction.
Verify that each statement is true for every positive integer n.
1(1!) + + n(n!) = (n + 1)! 1
Let
: 1(1!) + + n(n!) = (n + 1)! 1
Base step:
c
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 2
1. A menagerie consists of seven brown dogs, two black dogs, six gray cats, ten black
cats, ve blue birds, six yellow birds, and one black bird. Determine which of the
following statements are t
SEC 1104 / SWE 1104
(Mathematics For Computing 1)
TUTORIAL 1
1. Write the statement below in symbolic form using the symbols ~, and
and the indicated letters to represent component statements.
a) Juan is a math major but not a computer science major. (m=
Mathematics for Computing 1
SEC 1104 / SWE 1104
Name: Sharil Idzwan Shafie
(M. Sc. Quantitative Science)
Email: sharilidzwan.shafie@taylors.edu.my
Phone: ext. 5277
Learning Outcomes:
At the end of the course, students will be able to
Describe and di
UWE - Taylor's University
Learning Objective:
To apply the Principle of Mathematical Induction
To solve the Towers of Hanoi puzzle
To define a recurrence relation
UWE - Taylor's University
Mathematical Induction
Suppose we have a mathematical proposition,
LECTURE WEEK 3:
PROOF TECHNIQUES
LEARNING OBJECTIVES:
The student able to understand the structure of
mathematical proofs.
The student will be able to determine the direct
proofs of universal and existential statements.
The student will be able to disprov
Lecture 13
Learning Objectives:
Students will be able to determine the properties of trees
Students will be able to construct Minimum Spanning Trees
Students will be able to apply Kruskals and Prims Algorithms
Trees
A TREE is a connected graph and circuit
Lecture 12
Learning Objectives:
Students will be able to determine the properties of graphs.
Students will be able to construct Euler circuits
Students will be able to construct Hamiltonian circuits
Graphs
A graph G consists of two sets: set V of vertices
Lecture 11
Learning Objectives:
Students will be able to describe recurrence relation by Iterations
Student will be able to solve Second-Order Linear Homogeneous
with Constant Coefficients
Formulas to Simplify Solutions
obtained by Iterations
Example
Cons
Lecture 10
Students being able to understand the structure of Sequence
Students being able construct explicit formula for Recurrence
Relations
Student are able to apply Arithmetic and geometric progression
Sequence
Notation:
,
,
,
(read a sub k)
,
,
,
Lecture 9
To apply permutation and combination in solving problems.
To define the Binomial Theorem
To use and understand the relationship between Pascals Triangle,
Pascals Identity and the Binomial Theorem.
1
Activity 9.1
How many triangle can you draw us
Lecture 8: Counting
To apply Sum and product Rule in solving problems.
To apply Inclusion-exclusion Principle
To apply Pigeonhole Principle
Sum Rule
Everyday, there are 2 trains routine, 5 express
bus routine, and 4 flight routine from Malaysia to
Singapo
Lecture 7
Learning Objectives
To apply division algorithm
To apply the Euclidean algorithm
Algorithms
An algorithm is a systematic procedures (instructions)
for calculation.
Algorithms are basic to computer programs. Essentially,
a program implements one
Lecture 7
Learning Objectives
To apply division algorithm
To apply the Euclidean algorithm
Algorithms
An algorithm is a systematic procedures (instructions)
for calculation.
Algorithms are basic to computer programs. Essentially,
a program implements one
Lecture 6
Learning Objectives:
To describe functions
To define a one-to-one and onto function
To apply and distinguish special kinds of functions (i.e. absolute value, floor
and ceiling, logarithmic, exponential and polynomial)
Functions
Let X and Y be se
Lecture 5
Learning Objectives
To use set notations
To apply operations (union, intersection) on sets
To define de Morgans Laws for sets
To define relations on sets
To define set partitions
Set Theory
Set is a collection of objects
The cfw_ notation for s