Chapter 2: The Logic of Compound Statements
Section 2.3 Valid and Invalid Arguments
MH1300/MTH111 Lecture 4 (NTU)
2.3 Valid and Invalid Arguments
1 / 23
Synopsis
Todays objective: Get familiar with valid and invalid arguments.
Argument
Modus Ponens and Mo
Chapter 3: The Logic of Quantified Statements
Section 3.4 Arguments with Quantified Statements
MH1300/MTH111 Lecture 8 (NTU)
3.4 Arguments with Quantified Statements
1 / 17
Synopsis
Objective: We extend our knowledge on valid and invalid arguments in
Sect
Chapter 4: Elementary Number Theory and
Methods of Proof
Section 4.5 Direct Proof and Counterexample IV: Floor
and Ceiling
MH1300/MTH111 Lectures 11 (NTU)
4.5 Floor and Ceiling
1 / 37
Synopsis
Objective: More proof writing practices, connections between f
Chapter 4: Elementary Number Theory and
Methods of Proof
Section 4.2 Direct Proof and Counterexample II:
Rational Numbers
MH1300/MTH111 Lectures 9 (NTU)
4.2 Rational numbers
1 / 14
Synopsis
Objective: Practice proof writing through properties of rational
Chapter 1: Speaking Mathematically
Section 1.3 The Language of Relations and Functions
MH1300/MTH111 Lecture 2 (NTU)
1.3 Relations and Functions
1 / 13
Synopsis
Objective: Introduction to relations. Defining functions using relations.
Examples.
Definition
Chapter 3: The Logic of Quantified Statements
Section 3.2 Introduction to Predicates and Quantified
Statements II
MH1300/MTH111 Lecture 5 (NTU)
3.2 Predicates and Quantified statements II
1 / 31
Synopsis
Objective: Discuss statements involving quantifiers
Chapter 4: Elementary Number Theory and
Methods of Proof
Section 4.8 Application: Algorithm
MH1300/MTH111 Lectures 14 (NTU)
4.8 Application: Algorithm
1 / 14
Synopsis
Objective: We introduce two algorithms in elementary number theory. An
underlying tool i
Chapter 5: Sequences, Mathematical Induction,
and Recursion
Section 5.3 Mathematical Induction II
MH1300/MTH111 Lectures 13 (NTU)
5.3 Mathematical Induction II
1 / 26
Example 5.3.1
Proposition 5.3.1
For all integers n 1, 22n 1 is divisible by 3.
Proof (by
Chapter 6: Set Theory
Section 6.1 Set Theory: Definitions and the Element
Method of Proof
MH1300/MTH111 Lectures 15 (NTU)
6.1 Definitions and Element Method
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Synopsis
Objective: We review the basic definitions in Set Theory. Most of the
material in
MH1300/MTH111 Review
Objectives / Apologies
Proof writing improvements
Overview / Exam
Discounts
MH1300/MTH111 Lectures 22 (NTU)
Review
1/5
Objectives / Apologies
Recall: About this course
There are no big theorems nor difficult proofs in this course.
You
Chapter 8: Relations
Section 8.4 Modular Arithmetic
MH1300/MTH111 Lectures 20 (NTU)
8.4 Modular Arithmetic
1 / 26
Synopsis
Objective: Introduction to modular arithmetic. Pave the way to the proof
of the fundamental theorem of arithmetic.
Properties of Con
Chapter 8: Relations
Section 8.5 Partial Order Relations
MH1300/MTH111 Lectures 21 (NTU)
8.5 Partial Order Relations
1 / 17
Synopsis
Objective: We will only discuss partial order relations and total ordering,
and some examples to help us understand the de
Chapter 5: Sequences, Mathematical Induction,
and Recursion
Section 5.1 Sequences
MH1300/MTH111 Lectures 12 (NTU)
5.1 Sequences
1 / 27
Synopsis
Objective: A review of sequences and sums.
This is mostly a review of past knowledge of that you already
learnt
Chapter 6: Set Theory
Section 6.3 Disproofs and Algebraic Proofs
MH1300/MTH111 Lectures 16 (NTU)
6.3 Disproofs, Algebraic Proofs
1 / 11
Synopsis
Objective: Using Venn diagrams to help us find counterexamples to
disprove false set identities.
Prove set ide
Sections 1.11.6: Complex Numbers
(Complex Variables and Applications,
BrownChurchillVerhey)
Complex Numbers
MH1300/MTH111 Lecture 6 (NTU)
Complex Numbers
1 / 33
Synopsis
Objective: An introduction to complex numbers and their basic properties.
This is mos
Chapter 4: Elementary Number Theory and
Methods of Proof
Section 4.3 Direct Proof and Counterexample III:
Divisibility
MH1300/MTH111 Lectures 10 (NTU)
4.3 Divisibility
1 / 28
Synopsis
Objective: More proof writing practices via properties of divisibility.
MAS 111 FOUNDATION OF MATHEMATICS
ASSIGNMENT 9 HINTS
1. The following two binary relations are dened on the set A = cfw_0, 1, 2, 3. For each relation, determine whether it is reexive, symmetric, transitive. Give a counterexample in each case in which the
MAS 111 FOUNDATION OF MATHEMATICS
ASSIGNMENT 8 TUTORIAL DATES: 20, 21, 22/10/09
1. The following two binary relations are dened on the set A = cfw_0, 1, 2, 3. For each relation, determine whether it is reexive, symmetric, transitive. Give a counterexample
NANYANG TECHNOLOGICAL UNIVERSITY
MAS 111 FOUNDATION OF MATHEMATICS
Assignment 2 TUTORIAL TIME:31/08, 1, 2/09, 2009
In question 7, several properties of Fibonacci sequence are listed. Try to nd proofs by yourself. After proving these by mathematical induct
NANYANG TECHNOLOGICAL UNIVERSITY
MAS 111 FOUNDATION OF MATHEMATICS
ASSIGNMENT 1 Hints
1. Prove that the square of any odd integer leaves remainder 1 upon division by 8. Hint: Let m be an odd integer, and we can let m = 2k + 1, where k is an integer. Then
NANYANG TECHNOLOGICAL UNIVERSITY
MAS 111 FOUNDATION OF MATHEMATICS
ASSIGNMENT 1 TUTORIAL TIME: 17/08/09
1. Prove that the square of any odd integer leaves remainder 1 upon division by 8. 2. Prove that the product of two numbers of the form 4k + 3 is of th
MAS 111 FOUNDATION OF MATHEMATICS
PRACTICE EXERCISES FUNCTIONS
1. Does the formula f (x) =
x2
1 dene a function f : R R? 2
2. For each of the following functions, determine whether it is one-to-one and determine its range. (a) f : Z Z, f (x) = 2x + 1; (b)
MAS 111 FOUNDATION OF MATHEMATICS
ASSIGNMENT 12 HINTS
1. Multiply (a) (67i)(8+i), 2323 (b) ( + i)( i), 3232 1 1 (c) ( + i)(1 2i). 2 2
Hint: Apply the formula (a + bi)(c + di) = (ac bd) + (bc + ad)i. 2. Express z 1 in the form z 1 = a + bi for: (a) z = 6 +
NANYANG TECHNOLOGICAL UNIVERSITY
MAS 111 FOUNDATION OF MATHEMATICS
ASSIGNMENT 12 TUTORIAL DATES: 09, 10, 11/11/2009
1. Multiply (a) (6 7i)(8 + i), 2323 (b) ( + i)( i), 3232 1 1 (c) + i(1 2i). 2 2
2. Express z 1 in the form z 1 = a + bi for: (a) z = 6 + 8i
MAS 111 FOUNDATION OF MATHEMATICS
ASSIGNMENT 10 HINTS
1. Here are some relations on a set A = cfw_1, 2, 3, 4, 5. Determine for each relation whether it is reexive, whether it is transitive, whether is it antisymmetric. Draw the directed graph that represe