MA1100 Lecture 8
Mathematical Proofs
Congruences Proof by Cases Absolute Values
Chartrand: section 3.4, 4.2, 4.3
1
Congruence
Definition Let a, b and n be integers with n > 1. If n divides a b, we say that a is congruent to b modulo n Notation Example a
MA1100 Lecture 7
Mathematical Proofs
Parity of integers Divisibility of integers Direct Proofs Proof by Contrapositive
1
Chartrand: section 3.2, 3.3, 4.1
Announcement
Homework set 1 due today
Write your name/student number/tutorial group Hand in by end
MA1100 Lecture 7
Mathematical Proofs
Parity of integers Divisibility of integers Direct Proofs Proof by Contrapositive
1
Chartrand: section 3.2, 3.3, 4.1
True Statements
Mathematical statements that are true are known as mathematical facts. A propositi
MA1100 Lecture 6
Logic
Quantified statements
Chartrand: section 2.10
1
Announcement
Homework set 1 due next Tuesday
Do not mix up tutorial & HW problem sets Marking Scheme (next slide)
Math Clinic
http:/ww1.math.nus.edu.sg/clinic.htm Start next week
Virtu
MA1100 Lecture 6
Logic
Quantified statements
Chartrand: section 2.10
1
Quantified Statements
Examples 1)There are integers x and y such that 2x + 5y = 7 2)For all integers x and y, 2x + 5y = 7 When a quantifier is attached to an open sentence, the sentenc
MA1100 Lecture 5
Logic
Converse and contrapositive Biconditionals Tautologies and contradictions Logical equivalence
1
Chartrand: section 2.6 2.9
Announcement
Lecture quiz starts today
Sit with your buddy You need to key in both your student numbers usi
MA1100 Lecture 5
Logic
Converse and Contrapositive Biconditionals Tautologies and Contradictions Logical Equivalence
1
Chartrand: section 2.6 2.9
Summary (from last lecture)
Operator Negation Implication Conjunction Disjunction Standard form not P If P
MA1100 Lecture 4
Logic
Statements Open sentences Disjunction and conjunction Negation Implication
1
Chartrand: section 2.1 2.5
Announcement
Tutorial begin next week
Tutorial problems in Intro handouts page 4 Attendance will be taken
Homework set 1 due w
MA1100 Lecture 4
Logic
Statements Open sentences Disjunction and conjunction Negation Implication
1
Chartrand: section 2.1 2.5
Mathematical Language
Not exactly the same as English A dialect of English Expressing mathematical ideas precisely The grammar
MA1100 Lecture 3
Sets
Set Operations Indexed Collection of Sets Partitions of Sets Cartesian Products of Sets
1
Chartrand: section 1.3 1.6
Announcement
Tutorial balloting close tomorrow (NUS students only) Lecture quiz buddy (LQB) sign up form (for thos
MA1100 Lecture 3
Sets
Set Operations Indexed Collection of Sets Partitions of Sets Cartesian Products of Sets
1
Chartrand: section 1.3 1.6
Complement
Let A be a subset of a universal set U. The complement of A is the set of all elements of U that are no
MA1100 Lecture 9
Mathematical Proofs
Proof by Contradiction Existence Proof
Chartrand: section 5.2, 5.4
1
Proof by Contradiction
To prove statement R is true Assume ~R is true Try to get a contradiction Conclude that R must be true To prove universal st
This lecture will not be conducted physically at LT27 on Oct 9. It will only be available on IVLE in webcast format from Oct 9 onward.
MA1100 Lecture 15
Relations
Equivalence relations Equivalence classes Equivalence relations and partitions
Chartrand:
This lecture will not be conducted physically at LT27 on Oct 6. It will only be available on IVLE in webcast format from Oct 6 onward.
MA1100 Lecture 14
Relations
Representation of relation Domain and range of relation Reflexive, symmetric, transitive r
MA1100 Lecture 13
Revision Lecture
Justification Quantified Statements Proving Set Relations Using Hypothesis Without Loss of Generality Common mistakes
Chartrand: chapters 1 - 5
1
Announcement
Today: Collect back your HW2 Tomorrow: Online survey closed
MA1100 Lecture 13
Revision Lecture
Justification Quantified Statements Proving Set Relations Using Hypothesis Without Loss of Generality Common mistakes
Chartrand: chapters 1 - 5
1
Justify your answers
What are you expected to do? Give an explanation If
MA1100 Lecture 12
Mathematical Induction
Using PMI on other universal sets Strong PMI Variations of strong PMI
Chartrand: section 6.4
1
Announcement
Collection of clickers (No.1 to 150) Mid-term test: Oct 2 (12.00-1.30pm) at MPSH1. Handouts (please get
MA1100 Lecture 12
Mathematical Induction
Using PMI on other universal sets Strong PMI Variations of strong PMI
Chartrand: section 6.4
1
Other Universal Sets
Possible to apply PMI to prove: (" n Z) P(n) 1. P(0) is true 2. For all k 0, if P(k) is true, th
MA1100 Lecture 11
Mathematical Induction
Axiom of Induction Principle of Mathematical Induction Base Case Inductive Step
1
Chartrand: section 6.1, 6.2
Announcement
Hand in homework 2 (according to tutorial group). Write your name, matric number and tuto
MA1100 Lecture 11
Mathematical Induction
Axiom of Induction Principle of Mathematical Induction Base Case Inductive Step
1
Chartrand: section 6.1, 6.2
Example 1
Cardinality of power set Let A be a finite set with n element. How many elements does P(A) h
MA1100 Lecture 10
Mathematical Proofs
Proving involving
Sets relations Power sets Cartesian products Indexed collection of sets Empty sets
Chartrand: section 4.4, 4.5, 4.6
Proving A B
Example A, B are subsets of universal set Z B = cfw_ x Z | x is even A
MA1100 Lecture 9
Mathematical Proofs
Proof by Contradiction Existence Proof
Chartrand: section 5.2, 5.4
1
Announcement
Homework set 2 due next Tuesday Homework 1 solutions & scores in IVLE Lecture Quiz scores (last week) in IVLE
Lecture 9
2
Knights or K
MA1100 Lecture 2
Sets
Set Notations Set Relations Set Operations
Chartrand: section 1.1 1.3
1
Announcement
Chartrands book available at Co-op now. Tutorial balloting starts today (NUS students only) Introductory handouts Start looking for your lecture q
Lecture Lecture 13
3.5 Dimensions 3.6 Transition Matrices
Chapter 3
Vector Spaces
1
Announcement Mid-term test tomorrow
More past year/sample test papers
Lab session 3 next week
(Worksheets available)
Lecture 13
Announcement
2
A basis for Rn is not a basi
MA1100 Lecture 23
Cardinality
Equivalence of sets Infinite sets Denumerable sets Countable sets Uncountable sets
Chartrand: 10.1, 10.2, 10.3
Which set is larger?
Question: N (0,1) They are both infinite sets Can we compare their size?
Lecture 23
2
Equiva