MA1100 HW3
Answer key
Only questions that are boxed are checked in details by the graders; comments will be
made on the scripts to these question if necessary.
Each question in the problem set that is reasonably attempted will receive 3 marks;
incomplete
MA1100
Lecture 21
Number Theory
Congruence modulo
Modular arithmetic
Integers modulo n
Greatest common divisor
Euclidean Algorithm
Textbook: section 9.1 - 9.2
Modular Arithmetic
True or false
Let a Z.
1. If a 0 mod 6, then a2 0 mod 6.
2. If a2 0 mod
MA1100
Lecture 18
Functions
Composition of functions
Cardinality of infinite sets
Countable sets
Textbook: section 7.3 7.4
Announcement
Online Survey
Homework 4
Results in Workbin (Misc folder)
Follow up action
Problem set in workbin
due next Thursday
Enr
MA1100
Lecture 19
Relations
Defining relations
Congruence modulo
Reflexive, symmetric, transitive
Equivalence relation
Equivalence classes
Textbook: section 7.4, 8.1 8.3
Announcement
Homework
Typo in textbook
HW4 due next Thursday
HW3 to be returned next
MA1100
Lecture
Lect e 22
Number Th
N
b Theory
Linear Combination
Relatively Prime
l
l
Inverse modulo
Cryptography
RSA method
Textbook: Section 9.2 - 9.3
Ref: TB P354
Reversing Euclidean Al
ld
Algorithm
h
Example
a = 42 and b = 234
gcd(234, 42) = 6
d(
)
Fi
MA1100
Lecture 21
Number Theory
Congruence modulo
Modular arithmetic
Integers modulo n
Greatest common divisor
Euclidean Algorithm
Textbook: section 9.1 - 9.2
One month to final exam
HW4 will be returned to you in week 13
(during tutorial class)
Las
MA1100
Revision Lecture
Presented by A/Prof Victor Tan
Justification
Quantified Statements
Using Hypothesis
Without Loss of Generality
Common mistakes
Textbook: chapters 1 - 4
Justify your answers
What are you expected to do?
Give an explanation
If P ~Q
MA1100
Lecture 20
Relations/Number Theory
Equivalence classes
Equivalence relations and partitions
Congruence modulo n
Modular arithmetic
Textbook: section 8.3, 9.1
4 more
lectures
2 more
lecture
quizzes
3 more
tutorials
Lesson Plan (What lies ahead)
1 mo
National University of Singapore
Department of Mathematics
Semester 1, 2013/2014
MA1100
Tutorial Set 0
This is NOT your homework set, and you are not required to hand it in.
This tutorial set is for self practice and will NOT be discussed during tutorial
National University of Singapore
Department of Mathematics
Semester 1, 2013/2014
MA1100
Tutorial Set 1
IMPORTANT! This is NOT your homework set, and you are not required to hand it in.
However you should try your best to answer them before attending tutor
National University of Singapore
Department of Mathematics
Semester 1, 2013/2014
MA1100
Tutorial Set 1
IMPORTANT! This is NOT your homework set, and you are not required to hand it in.
However you should try your best to answer them before attending tutor
Tutorial Set 1 Solutions
1. Determine whether each of the following conditional statements is true or false:
(a) If 1 + 1 = 3, then 36 = 6.
1 + 1 = 3 is false. 36 = 6 is false. So the conditional statement is
true.
(b) If cos = 0, then sin = 0.
cos = 0 is
Tutorial Set 2 Solutions
1. Express each of the following statements in symbolic form using quantiers
or .
(a) Every real number is positive, negative or zero.
(b) The integer 13 is not a square.
(c) There is at least one real number whose square is 13.
National University of Singapore
Department of Mathematics
Semester 1, 2013/2014
MA1100
Tutorial Set 4
IMPORTANT! This is NOT your homework set
However you should try your best to answer them before attending tutorial classes.
This tutorial set will be di
National University of Singapore
Department of Mathematics
Semester 1, 2013/2014
MA1100
Tutorial Set 3
IMPORTANT! This tutorial set will be discussed online during Week 5 (Elearning week): Sep 9-13.
There will be no physical tutorial class for this tutori
MA1100
Lecture 20
Relations/Number Theory
Equivalence classes
Equivalence relations and partitions
Congruence modulo n
Modular arithmetic
Textbook: section 8.3, 9.1
Tutorial Group
A = the set of all MA1100 students
R = cfw_ (x,y) A A | x and y belong to t
MA1100
Lecture 19
Relations
Defining relations
Congruence modulo
Reflexive, symmetric, transitive
Equivalence relation
Equivalence classes
Textbook: section 7.4, 8.1 8.3
From Last lecture
How to show?
How to show an infinite set A is countable?
Construct
MA1100
Lecture 12
Mathematical Induction
Variation of PMI
Strong PMI
Variations of strong PMI
Well-Ordering Principle
Textbook: section 5.3
Other Universal Sets
Possible to apply PMI to prove:
(" n Z+) P(n)
(" n Z) P(n)
(" x Q+) P(x)
(" x Q) P(x)
What abo
MA1100
Lecture
Lect e 11
Mathematical Induction
Principle of Mathematical Induction
Basis Step
Inductive Step
Textbook: 5.1 5.2
Number of subsets
b
f b
Example
Let A be a finite set with n elements. How many
subsets does A have?
A = cfw_a
subsets: , cf
MA1100
Lecture 7
Proof
Theorem
Definition
Direct proof
Even and odd
Rational and irrational
Divisibility
Textbook: section 4.1 4.3
Announcement
Homework set 1 due this Thursday during lecture
Elearning Week
Whole of next week
No physical classes for lectu
MA1100
LECTURE 13 (REVISION)
Test Information
Summary of Proofs
Tutorial 5 Solutions
Things to bring for the test
Matric Card:
For identification purposes
Stationery:
Pen (no writing paper, pencil)
Helpsheet:
Prepare one handwritten A4 size helpsheet writ
MA1100
Lecture 11
Mathematical Induction
Principle of Mathematical Induction
Basis Step
Inductive Step
Textbook: 5.1 5.2
Announcement
Homework 2 (due this Thursday)
Term break next week
Mid-term Test in Week 7 (week after term break)
More Test Informat
MA1100
Lecture 14
Set Theory
Proving involving Set Relations
Element method
Algebra of Sets
Set operations
Set Identity
Textbook: section 6.1 6.3
Ref: TB P218
Proving A B
Example
A, B are subsets of universal set Z
A = cfw_ x Z | x = 8k 1 for some integer
MA1100
Lecture 15
Set Theory
Cartesian Product
Power Sets
Partitions of Sets
Indexed Collection of Sets
Textbook: section 6.1 6.2
Cardinality
The number of elements in a finite set A
is called the cardinality of A and is denoted by |A|.
Example
|cfw_a, b,
MA1100
Lecture
Lect e 16
Functions
F
ti
Images and inverse images
Range
Examples of functions
Injective functions
Textbook:Lecture 16
section 7.1 7.2
Ref: TB P256
Functions
A function from a set A to a set B
denoted by f: A B is a rule that
associate with
MA1100
Lecture 16
Functions
Images and inverse images
Range
Examples of functions
Injective functions
Textbook:Lecture 16
section 7.1 7.2
Ref: TB P256
Functions
A function from a set A to a set B
denoted by f: A B is a rule that
associate with every eleme