2.1 2.2 Part1 Preview Sets: Definitions and Venn diagram I couldn't finish all. We covered till p.8 the union & intersection of three sets. I will continue next week.
2.1 Exercises 17
2.2 Exercises 29
2.2 Exercises 5,6 (for #5 and #6, draw Venn diagrams to show the identities, as done in class. The solution given in the textbook is an advanced subject that we will learn in about a month later.)
2.1 2.2 Part1 Preview Sets: Definitions and Venn diagram Go through "Set builder notations" on pp.3-4 and p.8
2.1 2.2 Part1 Preview Sets: Definitions and Venn diagram Review exercises on pp.5-6
2.2 Exercises 21 (for #21a, draw Venn diagrams to show the identities, as done in class; for #21b, first, draw Venn diagrams to show the identities; second, use Table of Set Identities and via algebraic process, as done in class. The solution given in the textbook is an advanced subject that we will learn in about a month later.)
I will send you over email a little before the recitation starts.
A quiz will be held after the recitation, during the last 10 minutes of the class.
1.1 Propositions/Statements, Logical connectives, True-False table, started. Will continue next week.
No recitation or quiz due to Dr.King's birthday makeup (Monday class meet)
Do examples and exercises appeared in the lecture files <- important!
2.1 Exercises 1
2.2 Exercises 21 (Review how it was done in class 2.1 2.2 Part2 Preview Set Identities via Venn diagram)
For #21a, draw Venn diagrams to show the identities
for #21b, first, draw Venn diagrams to show the identities; second, use Table of Set Identities and via algebraic process.
2.1 Exercises 9,11,21,29,37
8.5 Exercises 1 <- Yes, it is Section 8.5. Oddly, our textbook doesn't have an appropriate 2.1-2.2 exercise problem asking on the topic. I picked one from this later section.
2.2 Exercises 53,57,59
1.1 Exercises 38a,c,e, 11a,b,c,h, 13a--d, 17a,b,d
Recitation 2 and Quiz 2
1.1 Propositions/Statements, Logical connectives, True-False table
2.2 Part 4 Proof of Set identities (Practicing Logical equivalences in 1.3),
Do examples and exercises appeared in the lecture files <- important!
1.1 Exercises 19, 17c,h,25,29
1.2 Exercises 7
1.3 Exercises 5,7,9,11
1.3 Logical equivalences Review "Example" and do "In-class exercise" on p.10 (1.3 Math/English, Logically equiv statements, and Negation) <-- newly added on 2/5
2.2 Exercises 21a,b, 20a, 33 (HInt: Contrapositives are logically equivalent; Also, as in Tables 7 & 8 Logically equiv. involving Conditional/Biconditional Statements) --> postpone to next week
Recitation 3 and Quiz 3
2.2 Part 4 Proof of Set identities (Practicing Logical equivalences in 1.3),
1.4 2.1 Part2 Univ Exis Quantifiers (one variable), T/F statements
Do examples and exercises appeared in the lecture files <- important!
2.2 Exercises; "Prove" in the followowing problems means "rigorous proof" done in class
#21 (see pp10-11 in 2.2 Part 4 Proof of Set identities (Practicing Logical equivalences in 1.3)), #20a
2.1 Exercise #47
Do as done in class; 1.4 2.1 Part1 Predicates (one variable), Truth sets
(i) First, write the truth set using the set building notation. Include the domain and write the property of the variable(s) in English and/or standard math, as opposed to symbols of predicates such as P(x).
(ii) Next, conclude the set specifying the element in a more concrete manner, such as roster method, well-known set notations, or intervals, etc
(iii) Do your best to give a reasonable Venn diagram
2.1 Exercise #45 ("truth value" means True/False)
Do as done in class; 1.4 2.1 Part2 Univ Exis Quantifiers (one variable), T/F statements
Justify your answer T/F. If a universal statement is false, give a counter-example.
Except for obvious cases, it is usually clearer if you do (i)--(iii) in finding the truth sets, then thinking of "all " or "exist ".
Showing (i)--(iii) in finding the truth set for a predicate with the given domain will serve as a "justification" for your answer.
1.4 Exercise #37
1.4 Exercise #11
Recitation 4 and Quiz 4
No homework this week as 2/20 will be an exam.
Midterm 1
1.4 Exercise #7, 23
1.4 Example on pp.6-7 in 1.4 2.1 Part4 Variabl-ize
1.4 Supplemental Problems #1 and #2 on pp.8-11 1.4 2.1 Part4 Variabl-ize (Note that Supp Prob #1 solution has been included in the lecture file)
1.4 Exercise #41
1.5 Exercise #45 Postpone to next week
Recitation 5 and Quiz 5
1.5 Exercises #25, 27a,b,g,i, 39, 45
Recitation 6 and Quiz 6
1.7 Pt1 Lecture video1 (the last a few seconds have no audio; stop and move to the next video below)
See the lecture files 1.5 Part1 Predicate (Multi-variables) Nested Universal/Existential Quantifiers pp. 4--7 to review
how to determine T/F for a mixed quatified statements and how to justify your answer.
Recall the statement "Every non-negative real number has an multiplicative inverse."
See 1.5 Part2 Compound Nested Quantifiers, Variabl-ize pp.2--3 to review
how to express it using quantifiers, predicates, and logical connectives;
how to 'prove' that the statement is true
1.7 Exercises #1, 2,3,4, 6,7
See the analysis and steps for 1.7 proof problems from lecture note 1.7 Part1 Direct Proofs. Your solutions must follow this procedure, as done in class.
I cover Section 1.6 only as a tool for 1.7-1.8 Proofs. Hence no homework from 1.6. However, do not ignore 1.6 covered in the videos. A few basics in 1.6 will help you follow 1.7 proofs.
Recitation 7 and Quiz 7
1.6 Part3 Rules of Inf Contrapositive Proofs
1.7 Part2 Proof by Contrapositive
1.7 Part3 Proof by Contradiction We did all except the Example 12 on pp.8-9. I will continue next week.
1.7 Part4 IFF, TFAE, Proof/Disproof, Fallacy postponed to next week
1.5 Exercises #31, 38
1.7 Exercises #17,13, 29,43 postponed to next week
1.7 Exercises #31 and Supplemental Problem #1: Prove or disprove the statement "For every integer n, if n is odd, then (n-1)/2 is odd."
Recitation 8 and Quiz 8
1.7 Part3 Proof by Contradiction finished.
1.7 Part4 IFF, TFAE, Proof/Disproof, Fallacy
IFF
Prove/Disprove (Proof for Ex15 is more appropriate to do after learning 1.8 Proof by Cases. So we will come back to Ex15 afterwards.)
Supp Prob #1 in 1.7 Exercises #31 and Supplemental Problem #1
Omit: TFAE (pp.2-3), Beginner's Mistakes in Proofs; Fallacy (p.5) - read them on your own
Questions in depth regarding the given statement and prove/disprove its converse using Ex12 in 1.7 Part4 IFF, TFAE, Proof/Disproof, Fallacy and Supp Prob #1 in 1.7 Exercises #31 and Supplemental Problem #1
When you take a theory math course, you have to think about this kind of question all the time.
You can omit detailed "logic behind" on pp.1--6. Instead, I recommend you understand "intuitive diagram" on p.7.
1.8 Proof with the notion of "Without Loss of Generality (WLOG)"
1.8 Proof involving Uniqueness postponed to next week
Several of homework from 1.8 below are EXAMPLES as some of the exercises are Review them.
1.7 EXAMPLE 11
1.7 Exercises 29
1.8 EXAMPLE 4, Exercise 9
1.8 EXAMPLE 13 postponed to next week
1.8 EXAMPLE 7
1.8 EXAMPLE 6, Exercise 33
1.7 EXAMPLE 15; though this example is in section 1.7, its proof is similar to (but easier than) 1.8 EXAMPLE 6.
Reading assignment: Beginner's Mistakes in Proofs; Fallacy on p.5 in 1.7 Part4 IFF, TFAE, Proof/Disproof, Fallacy
Recitation 9 and Quiz 9
The last two pages of 2.4 Sequence and Series
Revisit "Proof by Cases/WLOG":
Dividing good cases are often the most difficult task in 'proof by cases/WLOG'. Hence we will practice again this week. In addition to reviewing the homework from the last week on the topic, do
1.8 Exercises 5
Review Quiz 9 (Solutions have been emailed to you.)
1.8 EXAMPLE 13
2.4 Exercises 1, 13
2.4 Redo Examples and Exercises done in 2.4 Sequence and Series EXCEPT #17f on p.14 (Theoretically, you can solve this problem as we learned everything you need to solve the problem. In practice, it might be be difficult and I will do this problem next week.)
"Redo" means, first review & understand what was done in class, then attempt to solve them without looking at the solutions.
2.4 EXAMPLE 22
Recitation 10 and Quiz 10
Discussing Recitation 10, in particular 2.4 Exercises 16c,d, #26d; the final answer should be obtained by a proper process (for instance by displaying & recognizing a well-known series that appears in the process), not by guessing
No homework this week as 4/18 will be an exam.
Midterm 2
5.1 Mathematical Induction Omit from pp.28 onwards - Induction and Increasing/Decreasing sequences from Calculus II; Induction and Set identities
4.1 Divisibility and 5.1 Divisibility and Induction Omit from pp.8 onwards - Divisibility and Induction
4.1 Modular Arithmetic moved to next week
5.1 EXAMPLE 3 (solution can be found on p.21 5.1 Mathematical Induction)
5.1 Exercises 5,7,24
4.1 Exercises 7,8
4.1 Additional problems in 4.1 Divisibility and 5.1 Divisibility and Induction
pp.5-6 #2
p.7 Prove Theorem (Relations in congruence mod m) (a)--(c)
Recitation 11 and Quiz 11
We omit most contents.
We cover only pp.8--14 : "Definition of congruence modulo m" and "Theorem of remainder module m", and then p.26 "Three relations in Modular Arithmetic".
Note that pp.8--14 are overlaying slides and are basically one page.
The actual contents of the definition and the theorems have been already covered on pp.5--7 in 4.1 Divisibility and 5.1 Divisibility and Induction
We will need the notation and the theorem in 9.1 Relations and 9.5 Equivalence Relation
9.5 Equivalence Relation <-- I will finish "equivalence classes" in the beginning of the 5/1 class before starting recitation & quiz.
Equivalence Relation (pp.1-9)
Equivalence Classes (pp.10-18)
4.1 Three relations in Modular Arithmetic
p.26 in 4.1 Modular Arithmetic #Prove "4.1 HW/Rec Three relations in Modular Arithmetic"
9.1 Relation
9.1 Exercises #1 (warm-up), #7a,d,e (done in 9.1 Relations; Do only three properties, reflexive, symmetric, transitive)
9.5 Equivalence Relation
9.5 Exercises #1 (warm-up), #11 (done in 9.5 Equivalence Relation ), #13, #15(optional)
Do the preample questions, as below, as done in 9.5 Equivalence Relation
a) the ground set A (required to write on recitation/quiz/exam)
b) rewriting aRb (required to write on recitation/quiz/exam)
c) side heuristic work (not required to write but you do need this step for your own understanding)
p.5 in 9.5 Equivalence Relation #Prove "9.5 HW/Rec Congruence Modulo m is an Equivalence Relation"
9.5 Equivalence Classes <-- You can postpone till tomorrow.
p.16 in 9.5 Equivalence Relation #Answer "9.5 HW/Rec Supplemental Ex Modular Equivalence Classes"
9.5 Exercises #30 (done in class 9.5 Equivalence Relation), #32