Calendar for 2022 - 2023 Fall Semester MATH 3070/5070 Number Theory, taught by Mark Barsamian

Week 1 (Mon Aug 22 through Fri Aug 26)

Mon Aug 22:

• Section to Discuss: 1.1 Variables
• Homework: H01.1: 1.1#2,6,11,13

Tue Aug 24:

• Section to Discuss: 2.1 Logical form and Logical Equivalence
• Homework: H02.1: 2.1#8,22,26,28,33,34,42,45 ( video ) ( notes )

Fri Aug 26:

• Section to Discuss: 2.2 Conditional Statements
• Homework: H02.2: 2.2#8,15,{20b,22b,23b},21,33,35,41,43 ( video ) ( notes )
• Class Presentations for Fri Aug 26
  • Gretchen Angst CP1: Which of these statements is true? Explain
    1. If the moon is made of green cheese, then our class meets in Morton Hall.
    2. If the moon is made of green cheese, then the world is flat.
  • Evan Brooks CP1: Use a truth table to prove that \(p \rightarrow q\) is logically equivalent to \( (\sim p) \vee q\).
  • Michael Cooney CP1:
    1. Given that \(p \rightarrow q \equiv (\sim p) \vee q\), use DeMorgan to find \( \sim (p \rightarrow q) \).
    2. Negate statement S: If the truck is a Ford then the motorcycle is a Honda.

Week 2 (Mon Aug 29 through Fri Sep 2)

Mon Aug 29:

• Section to Discuss: 2.3 Valid and Invalid Arguments (Argument Forms)
• Homework: H02.3: 2.3#9,15,23,29,31,32 ( video ) ( notes )
• Quiz Q01 on Mon Aug 29 Covering Sections 2.1 and 2.2
  • 20 Minutes at the end of class
  • Similar to Suggested Exercises from Sections 2.1 and 2.2.

Wed Aug 31:

• Section to Discuss: 3.1 Predicates and Quantified Statements I
• Homework: Homework H03.1: 3.1#4,5,10,16,18,22,29 ( video ) ( notes )

Fri Sep 2:

• Section to Discuss: 3.2 Predicates and Quantified Statements II
• Homework: H03.2: 3.2#4,10,15,17,25,27,38,44 ( video ) ( notes )
• Class Presentations for Fri Sep 2 Show/Hide Presentations

  Julie Fausnaugh CP1:

  1. Let A be the statement "Every vehicle in the Morton Hall parking lot is a Ford". What is ~ A ?
  2. More generally, let A be the universal statement $$\forall x \in D \left( Q(x) \right)$$ What is ~ A ?

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  Lucas Fernandes CP1:

  1. Let B be the statement "There exists a vehicle in the Morton Hall parking lot that is a Ferrari". What is ~ B ?
  2. More generally, let B be the existential statement $$\exists x \in D \left( Q(x) \right)$$ What is ~ B ?

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  Luke Haskell CP1:

  1. Let C be the universal conditional statement $$\forall x \in D \left( IF \ \ P(x) \ \ THEN \ \ Q(x) \right)$$ What is the negation, ~ C ?
  2. As a specific example, let C be the following universal conditional statement: $$\forall x \in \mathbb{R} \left( x \lt 7 \rightarrow x^2 \lt 49\right)$$ Find the negation ~ C .
  3. For the specific example given, which is true, C or ~ C ? Explain.

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  Ethan Levingston CP1: Consider again the universal conditional statement C that Luke discussed earlier: $$\forall x \in \mathbb{R} \left( x \lt 7 \rightarrow x^2 \lt 49\right)$$

  1. Write the converse, contrapositive, and inverse of C
  2. Consider the four statements:
     • original statement C
     • converse of C
     • contrapositve of C
     • inverse of C
     Which are true and which are false? Explain.

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  Jennifer Lewis CP1: Let S be the following universal conditional statement: $$\forall n \in \mathbb{Z} \left(IF \ \frac{6}{n} \ is \ an \ integer, \ THEN \ \ n=2 \ or \ n=3\right)$$

  1. Write the converse, contrapositive, inverse, and negation of S
  2. Consider the five statments:
     • original statement S
     • converse of S
     • contrapositve of S
     • inverse of S
     • negation of S
     Which are true and which are false? Explain.
  Hide Fri Sep 2 Presentations

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Week 3 (Mon Sep 5 through Fri Sep 9)

Mon Sep 5 is Labor Day Holiday: No Class

Wed Sep 7:

• Section to Discuss: 3.3 Statements with Multiple Quantifiers
• Homework: H03.3: 3.3#2,3,6,17,19,20,23,26,30,38 ( video ) ( notes )
• Class Presentations for Wed Sep 7 Show/Hide Presentations

  Daniel Lipec CP1: Changing the order of multiple quantifiers.

  Statement B is obtained by switching the order the quantifiers of Statement A . $$\text{Statement A: }\forall x \in \left( \exists y \in \mathbb{Z} \left(x+y=0\right) \right) $$ $$\text{Statement B: } \exists y \in \mathbb{Z} \left( \forall x \in \mathbb{Z} \left(x+y=0\right) \right) $$

  Answer these questions.

  1. Which of the statements are true ? (one or both ) Explain.
  2. One of the two statments is famous property of the Integers. Which statement? What is the name of the property? Explain.
  3. Find the negation of any of any of the statements that are false.

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  Bradey LounsburyCP1: Changing the domain in quantifiers.

  Consider the following Statement S ..

  $$\text{Statement S: }\forall x \in D \left( \exists y \in D \left( xy=1 \right) \right) $$
  1. Write the negation for S .
  2. Is Statement S true when \(D = \mathbb{Z}\)? Explain.
  3. Is Statement S true when \(D = \mathbb{R}\) ? Explain.
  4. Is Statement S true when \(D = \mathbb{R}^+\) ? Explain.

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  Bryce NicholsonCP1: Interchanging \( \forall \) and \( \exists \) in multiple quantifiers.

  Consider Statement A and Statement B , which is obtained by interchanging \( \forall \) and \( \exists \) in the quantifiers of Statement A .

  $$\text{Statement A: }\forall x \in \mathbb{Z} \left( \exists y \in \mathbb{Z} \left( x \lt y \right) \right) $$ $$\text{Statement B: }\exists x \in \mathbb{Z} \left( \forall y \in \mathbb{Z} \left( x \lt y \right) \right) $$

  Is either of these statements true ? Explain.

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  Brittany Poss CP1: Interchanging x and y in the quantifiers.

  Consider Statement A and Statement B , which is obtained by interchanging x and y in the quantifiers of Statement A .

  $$\text{Statement A: }\forall x \in D \left( \exists y \in D \left( y=3x+2 \right) \right) $$ $$\text{Statement B: }\forall y \in D \left( \exists x \in D \left( y=3x+2 \right) \right) $$

  Answer the following questions.

  • Let the domain \(D\) be the set \(\mathbb{R}\). Is either of the statements A , B true ? Explain.
  • Let the domain \(D\) be the set \(\mathbb{Z}\). Is either of the statements A , B true ? Explain.

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  Clair Schmidt CP1: Choosing correct order for symbols to create a true statement.

  You find thirteen cards scattered on the ground. They have the following symbols on them:

  $$\boxed{\mathbb{R}} \ \ \boxed{\mathbb{Z}^+} \ \ \boxed{x} \ \ \boxed{y} \ \ \boxed{y= \sqrt{x}} \ \ \boxed{\forall} \ \ \boxed{\exists} \ \ \boxed{\in} \ \ \boxed{\in} \ \ \boxed{(} \ \ \boxed{(} \ \ \boxed{)} \ \ \boxed{)}$$

  Assemble the cards in an order that makes a correct statement. Explain.

  Hide Wed Sep 7 Presentations

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• Quiz Q02 on Wed Sep 7 Covering Sections 2.3, 3.1, and 3.2
  • 20 Minutes at the end of class
  • Similar to Suggested Exercises from Sections 2.3 and 3.2.

Fri Sep 9:

• Section to Discuss: 3.4 Arguments with Quantified Statements
• Homework: H03.4: 3.4#4,11,13,15,17,18,20,22,26
• Class Presentations: tba

Week 4 (Mon Sep 12 through Fri Sep 16)

Exam X1 on Mon Sep 12 Covering Chapters 1,2,3 Show/Hide Exam Info

Wed Sep 14:

• Section to Discuss: Epp 4th Edition Section 4.1 Direct Proof and Counterexample I: Introduction:
  • Exercises from Epp 4th Edition Section 4.1: # 4, 11, 16, 20, 22, 24, 25, 29, 31, 33, 35, 43, 44, 45, 50, 51, 52, 53, 54, 56, 57, 58
  • Video for Epp 5th Edition Section 4.1 = Epp 4th Edition Section 4.1: ( Video ) ( Notes )
  • Video for Epp 5th Edition Section 4.2 = more topics from Epp 4th Edition Section 4.1: ( Video ) ( Notes )
• Class Presentations for Wed Sep 14 Show/Hide Presentations
  • Jake Schneider CP1:
    1. Is 0 even ? Explain
    2. Can negative numbers be even ? Can they be odd ? Explain.
    3. Are even & odd opposites? That word, opposites , is not really a defined term in our course. A better question would be, are even & odd the negation of one another? That is, is every integer either even or odd? Explain.
    4. Assume that \(r\) and \(s\) are particular integers (a) is \(4rs\) even ? (b) is \(6r+4s^2+3\) odd? Explain.
  • Roman Simkins CP1: Prove or disprove: \(\forall n \in \mathbb{Z} \left(\text{If } n \text{ is odd then } \frac{n-1}{2} \text{ is odd.}\right) \)
  Hide Wed Sep 14 Presentations

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Fri Sep 16:

• Section to Discuss: Epp 4th Edition Section 4.1 Direct Proof and Counterexample I: Introduction:
  • Exercises from Epp 4th Edition Section 4.1: # 4, 11, 16, 20, 22, 24, 25, 29, 31, 33, 35, 43, 44, 45, 50, 51, 52, 53, 54, 56, 57, 58
  • Video for Epp 5th Edition Section 4.1 = Epp 4th Edition Section 4.1: ( Video ) ( Notes )
  • Video for Epp 5th Edition Section 4.2 = more topics from Epp 4th Edition Section 4.1: ( Video ) ( Notes )
• Class Presentations for Fri Sep 16 Show/Hide Presentations
  • Grace Smith CP1:
    1. Is 1 prime ? Explain.
    2. Can negative numbers be prime? Can they be composite? Explain.
    3. Are prime & composite opposites? That word, opposites , is not really a defined term in our course. A better question would be, are prime & composite the negation of one another? That is, is every integer either prime or composite? Explain.
    4. If \(r\) and \(s\) are positive integers, is \(r^2+2rs+s^2\) prime or composite or neither? Explain.
  • Rashid Al Busa'idi CP1: Prove or disprove the following statements.
    1. \(\exists n \in \mathbb{Z} \left(2n^2-5n+2 \text{ is prime.}\right) \)
    2. \(\forall n \in \mathbb{Z} \left(2n^2-5n+2 \text{ is composite.}\right) \)
    3. \(\forall n \in \mathbb{Z} \left(2n^2-5n+2 \text{ is prime OR } 2n^2-5n+2 \text{ is composite.}\right) \)
  • Nicole Strang CP1: Consider Statement \(S\): \(\forall n \in D\left(n^2-n+11 \text{ is prime.}\right) \)
    1. Let \(D=\{1,2,3,4,5,6,7,8,9,10\}\). Prove that \(S\) is true using the method of exhaustion .
    2. Now let \(D=\mathbb{Z}^+\). Is \(S\) true or false? Explain.
  Hide Fri Sep 16 Presentations

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Week 5 (Mon Sep 19 through Fri Sep 23)

Mon Sep 19:

• Section to Discuss: Epp 4th Edition Section 4.1 Direct Proof and Counterexample I: Introduction:
  • Exercises from Epp 4th Edition Section 4.1: # 4, 11, 16, 20, 22, 24, 25, 29, 31, 33, 35, 43, 44, 45, 50, 51, 52, 53, 54, 56, 57, 58
  • Video for Epp 5th Edition Section 4.1 = Epp 4th Edition Section 4.1: ( Video ) ( Notes )
  • Video for Epp 5th Edition Section 4.2 = more topics from Epp 4th Edition Section 4.1: ( Video ) ( Notes )
• Class Presentations for Mon Sep 19: Show/Hide Presentations
  • Gretchen Angst CP2: (Exercise 4.1#43 from Epp 4th Edition) Prove or disprove: The product of any two odd integers is odd.
  • Evan Brooks CP2: (Exercise 4.1#45 from Epp 4th Edition) Prove or disprove: The difference of any two odd integers is odd.
  • Michael Cooney CP2: (Exercse 4.1#50 from Epp 4th Edition) Prove or disprove: For all integers \(m,n\) if \(m-n\) is even, then \(m^3-n^3\) is even.
  • Julie Fausnaugh CP2: (Exercse 4.1#51 from Epp 4th Edition) Prove or disprove: For all integers \(n\) if \(n\) is prime, then \((-1)^n=-1\).
  • Lucas Fernandes CP2: (Exercse 4.1#52 from Epp 4th Edition) Prove or disprove: For all integers \(m\) if \(m>2\), then \(m^2-4\) is composite .
  Hide Mon Sep 19 Presentations

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Wed Sep 21:

• Section to Discuss: Epp 4th Edition Section 4.2 Direct Proof and Counterexample II: Rational Numbers:
  • Exercises from Epp 4th Edition Section 4.2: # 2, 4, 6, 9, 14, 15, 16, 17, 18, 19, 20, 21, 28, 31, 33
  • Video: Video for Epp 5th Edition Section 4.3 = Topics from Epp 4th Edition Section 4.2 ( link to video ) ( Link to Notes )
• Class Presentations for Wed Sep 21 Show/Hide Presentations
  
  
  • Luke Haskell CP2: The following statement about rational numbers is true . (It is the subject of Epp 4th Edition Exercise 4.2#15.)
    The product of any two rational numbers is a rational number.
    But the following statement irrational numbers is false .
    The product of any two irrational numbers is an irrational number.
    Prove that the statement about irrational numbers is false . (Give a counterexample.)
  
  
  • Ethan Levingston CP2: The following statement about rational numbers is true . (It is the subject of Epp 4th Edition Exercise 4.2#17.)
    The difference of any two rational numbers is a rational number.
    But the following statement irrational numbers is false .
    The difference of any two irrational numbers is an irrational number.
    Prove that the statement about irrational numbers is false . (Give a counterexample.)
  
  
  • Jennifer Lewis CP2: The following statement about rational numbers is true . (It is the subject of Epp 4th Edition Exercise 4.2#18.)
    If \(r\) and \(s\) are rational numbers, then \(\frac{r+s}{2}\) is a rational number .
    But the following statement irrational numbers is false .
    If \(r\) and \(s\) are irrational numbers, then \(\frac{r+s}{2}\) is an irrational number.
    Prove that the statement about irrational numbers is false . (Give a counterexample.)
  
  Hide Wed Sep 21 Presentations

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• Quiz Q03 on Wed Sep 21, covers Section 4.1

Fri Sep 23:

• Section to Discuss: Epp 4th Edition Section 4.2 Direct Proof and Counterexample II: Rational Numbers:
  • Exercises from Epp 4th Edition Section 4.2: # 2, 4, 6, 9, 14, 15, 16, 17, 18, 19, 20, 21, 28, 31, 33
  • Video: Video for Epp 5th Edition Section 4.3 = Topics from Epp 4th Edition Section 4.2 ( link to video ) ( Link to Notes )
• Class Presentations for Fri Sep 23 Show/Hide Presentations
  • Daniel Lipec CP02: Consider the list of expressions below. Which represent real numbers? Which represent rational numbers. Explain. (If you say that an expression represents a rational number, then you should give the ratio of integers that equals that expression.)
    1. 7
    2. -5
    3. 0
    4. 0/5
    5. 5/0
    6. 753.86234
    7. 753.86234234234� (Decimal repeats starting in the 3rd decimal place.)
  • Bradey Lounsbury CP02:
    1. What is the Zero Product Property ?
    2. Write the Zero Product Property as a universal conditional statement. Then write the contrapositive of that universal conditional statement.
    3. Why is the Zero Product Property introduced in Epp 4th Edition Section 4.2 on Rational Numbers? Give an example of how the Zero Product Property is used in Section 4.2.
  • Bryce Nicholson CP02: Consider the statement "The square of any rational number is rational"
    1. Rewrite the statement more clearly using quantifiers.
    2. Prove or disprove the statement.
  • Brittany Poss CP02: Consider the statement "The product of any two rational numbers is rational"
    1. Rewrite the statement more clearly using quantifiers.
    2. Prove or disprove the statement.
  • Clair Schmidt CP02: Consider the statement "The quotient of any two rational numbers is rational"
    1. Rewrite the statement more clearly using quantifiers.
    2. Prove or disprove the statement.
  
  Hide Fri Sep 23 Presentations

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Week 6 (Mon Sep 26 through Fri Sep 30)

Mon Sep 26:

• Section to Discuss: Epp 4th Edition Section 4.3 Direct Proof and Counterexample III: Divibility
  • Homework: Problems from Epp 4th Edition Section 4.3 # 2,3,4,12,15,16,19,20,24,26,27,28,29,30,32,36,37,38,41,47,48
  • Video: Video for Epp 5th Edition Section 4.4 = Topics from Epp 4th Edition Section 4.3 ( link to video ) ( Link to Notes )
• Class Presentations for Mon Sep 26 Show/Hide Presentations
  • Jake Schneider CP02:
    1. Do the symbols 5 / 7 and 5 | 7 mean the same thing ? Explain.
    2. Do the symbols 2 / 6 and 2 | 6 mean the same thing ? Explain.
    3. Frick and Frack are arguing. Frick says that 3 / 0 is undefined. Frack says that 3 | 0 is true. Who is right ? Explain.
    4. Donkey and Elephant are arguing. Donkey says that 0 / 5 is zero. Elephant says that 0 | 5 is false. Who is right ? Explain.
  • Roman Simkins CP02: Consider the statement "For all integers a , b , and c , if a | b and a | c , then a | b-c ."
    1. Write the statement formally, using symbols for the quantifier.
    2. Write the negation of the statement formally, using symbols for the quantifier.
    3. Prove or disprove the statement.
  • Grace Smith CP02: Consider the statement "For all integers a , b , and c , if ab | c then a | c and b | c."
    1. Write the statement formally, using symbols for the quantifier.
    2. Write the negation of the statement formally, using symbols for the quantifier.
    3. Prove or disprove the statement.
  • Nicole Strang CP02: Consider the statement "For all integers a , b , and c , if a | bc then a | b or a | c."
    1. Write the statement formally, using symbols for the quantifier.
    2. Write the negation of the statement formally, using symbols for the quantifier.
    3. Prove or disprove the statement.
  • Rashid Al Busa�idi CP02: Consider the statement "For all integers a , b , and c , if a | b then a ² | b ² ."
    1. Write the statement formally, using symbols for the quantifier.
    2. Write the negation of the statement formally, using symbols for the quantifier.
    3. Prove or disprove the statement.
  
  Hide Mon Sep 26 Presentations

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Wed Sep 28:

• Section to Discuss: Epp 4th Edition Section 4.3 Direct Proof and Counterexample III: Divibility
  • Homework: Problems from Epp 4th Edition Section 4.3 # 2,3,4,12,15,16,19,20,24,26,27,28,29,30,32,36,37,38,41,47,48
  • Video: Video for Epp 5th Edition Section 4.4 = Topics from Epp 4th Edition Section 4.3 ( link to video ) ( Link to Notes )
• Class Presentations for Wed Sep 28 Show/Hide Presentations
  • Gretchen Angst CP03:
    1. Suppose that a number \(a\) can be written in standard factored form $$a=p_1^{e_1}p_2^{e_2}\cdot\cdot\cdotp_k^{e_k}$$ where
       • \(k\) is a positive integer.
       • \(p_1,p_2, ...,p_k\) are prime numbers with \(p_1 \lt p_2 \lt ... \lt p_k\).
       • \(e_1,e_2, ...,e_k\) are positive integers.
       What is the standard factored form for \(a^3\)?
    2. Find the least positive integer such that $$2^4\cdot3^5\cdot7\cdot11^2\cdot k$$ is a perfect cube. Write the resulting product as a perfect cube.
  • Evan Brooks CP03: How many zeros are at the end of the number \(35^{113}\cdot 48^{23}\)? Explain how you can answer this question without actually computing the number. (Hint: \(10=2\cdot5\).)
  • Michael Cooney CP03: Prove that for any nonnegative integer \(n\), if the sum of the digits of \(n\) is divisible by \(9\), then \(n\) is divisible by \(9\). (Hint: Study Exercise 4.3#47 for a concrete example, and then generalize that example.)
  
  Hide Wed Sep 28 Presentations

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• Quiz Q04 on Wed Sep 28, Covers Epp 4th Edition Sections 4.2 and 4.3

Fri Sep 30 is Fall Break: No Class. Stay home and study math!!

Week 7 (Mon Oct 3 through Fri Oct 7)

Mon Oct 3:

• Section to Discuss: Epp 4th Edition Section 4.4 Direct Proof and Counterexample V: Division into Cases and the Quotient Remainder Theorem:
  • Exercises from Epp 4th Edition Section 4.4 Direct Proof and Counterexample V:: # 1,3,5,7,10,13,14,17,20,23,25,27,28a,29,9036,37,38,39
  • Video: Video for Epp 5th Edition Section 4.5 = Topics from Epp 4th Edition Section 4.4 ( link to video ) ( Link to Notes )
• Class Presentations for Mon Oct 3 Show/Hide Presentations
  • Julie Fausnaugh CP3: (similar to Epp 4th Edition Section 4.4 Exercises #7,10)
    1. Find \(50\text{ div }7\) and \(50\text{ mod }7\). Show the corresponding \(n=dq+r\) equation.
    2. Find \(-50\text{ div }7\) and \(-50\text{ mod }7\). Show the corresponding \(n=dq+r\) equation.
    3. Find \(56\text{ div }7\) and \(56\text{ mod }7\). Show the corresponding \(n=dq+r\) equation.
  • Lucas Fernandes CP3: (similar to Epp 4th Edition Section 4.4 Exercise #20)
    If \(c\) is an integer such that \(c \text{ mod } 13 = 5\), then what is \(6c \text{ mod } 13\)?
    In other words, if division of \(c\) by 13 gives a remainder of \(5\), what is the remainder when \(6c\) is divided by \(13\)? Explain using \(n=dq+r\) equations.
  • Luke Haskell CP3: (similar to Epp 4th Edition Section 4.4 Exercise #23)
    Prove that for all integers \(a\) and \(b\), if \(a \text{ mod } 7 = 5\) and \(b \text{ mod } 7 = 6\), then \(ab \text{ mod } 7 = 2\).
    Hint: Prove using \(n=dq+r\) equations.
  
  Hide Mon Oct 3 Presentations

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Wed Oct 5:

• Section to Discuss: Epp 4th Edition Section 4.4 Direct Proof and Counterexample V: Division into Cases and the Quotient Remainder Theorem:
  • Exercises from Epp 4th Edition Section 4.4 Direct Proof and Counterexample V:: # 1,3,5,7,10,13,14,17,20,23,25,27,28a,29,9036,37,38,39
  • Video: Video for Epp 5th Edition Section 4.5 = Topics from Epp 4th Edition Section 4.4 ( link to video ) ( Link to Notes )
• Class Presentations for Wed Oct 5 Show/Hide Presentations
  • Ethan Levingston CP3: (similar to Epp 4th Edition Section 4.4 Exercise #29) Suppose that \(n\) is an integer.
    1. What does the Quotient Remainder Theorem with \(d=2\) tell us about \(n\)?
    2. What does the Quotient Remainder Theorem with \(d=3\) tell us about \(n\)?
    3. Use the Quotient Remainder Theorem with \(d=3\) to prove that the cube of any integer has the form \(3k\) or \(3k+1\) or \(3k+2\) for some integer \(k\).
  • Jennifer Lewis CP3: (similar to Epp 4th Edition Section 4.4 Exercise #30) Recall that two consecutive integers can always be written in the form \(n(n+1)\), where \(n\) is an integer.
    1. What does the Quotient Remainder Theorem with \(d=3\) tell us about \(n\)?
    2. Your result of (a) should give you three possible cases for \(n\). What are the resulting values of \(n(n+1)\) in those three cases?
    3. Use the Quotient Remainder Theorem with \(d=3\) to prove that the product of any two consecutive integers has the form \(3k\) or \(3k+2\) for some integer \(k\).
  
  Hide Wed Oct 5 Presentations

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• Quiz Q05 on Wed Oct 5 over \(n=dq+r\) and MOD and DIV from Epp 4th Edition Section 4.4.

Fri Oct 7:

• Section to Discuss: Epp 4th Edition Section 4.6: Indirect Argument: Contraposition and Contraposition
  • Exercises:
    • Exercises about rational and irrational numbers: Epp 4th Edition Section 4.6 # 2 and these Three Extra Questions:
      1. Suppose that \( q=a/b \) is a rational number. What does that tell you about \(a\) and \(b\)?
      2. Suppose that \( q=a/b \) is a rational number and \(q=0\). What does that tell you about \(a\) and \(b\)?
      3. Suppose that \( q=a/b \) is a rational number and \(q \neq 0\). What does that tell you about \(a\) and \(b\)?
    • Exercises to be proven directly, not using contradiction and not proving the contrapositive: Epp 4th Edition Section 4.6 # 4,5,6,7,17
    • Exercises to be proven indirectly, by proving the contrapositive: Epp 4th Edition Section 4.6 # 10,12,14,15,19,21,22,24,25,26,27,28
    • Exercises to be proven indirectly by contradiction: Epp 4th Edition Section 4.6 # 16
    • Remark: Observe that out of all of the exercises that I assign in this section, only one needs to be proven by contradiction!
  • Video: Video for Epp 5th Edition Section 4.7 = Topics from Epp 4th Edition Section 4.6 ( link to video )
  • Notes: Notes from Video for Epp 5th Edition Section 4.7 ( Link to Notes )
  • Handouts: Handout on Overuse of the Method of Contradiction ( Link to Handout )
• Class Presentations for Fri Oct 7 Show/Hide Presentations

  Daniel Lipec CP3: Suppose that \(k\) is an integer.

  1. What does the Quotient Remainder Theorem (QRT) with \(d=3\) tell you about \(k\)?
  2. What does the remainder \(r\) have to be if the unknown \(k\) is going to be divisible by \(3\)?
  3. Suppose that it is known that \(k=3m+2\) for some integer \(m\). Is \(k\) divisible by \(3\)?

  Bradey Lounsbury CP3: Let \(S\) be the statement $$\forall m\in \mathbb{Z} \left( \exists n \in \mathbb{Z} \left( n \gt n \right) \right)$$

  1. Prove \(S\).
  2. Find the negation of \(S\).
  3. Is the negation of \(S\) true or false? Explain.

  Bryce Nicholson CP3: Let \(S\) be the statement $$\forall x \in \mathbb{R} \left( \text{ If }\sqrt{x} \in \mathbb{Q} \text{ then }x\in \mathbb{Q} \right)$$

  1. Prove \(S\).
  2. Find the contrapositive of \(S\).
  3. Is the contrapositive of \(S\) true or false? Explain.
  
  Hide Fri Oct 7 Presentations

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Week 8 (Mon Oct 10 through Fri Oct 14)

Bonus Quiz BQ1 due at the start of class on Mon Oct 10. ( Link to Bonus Quiz ) If you do not have a printout of this quiz, then you must find a way to print one. I will not accept hand-written solutions that are not on a printed quiz.

Mon Oct 10:

• Section to Discuss: Epp 4th Edition Section 4.6: Indirect Argument: Contraposition and Contraposition
  • Exercises:
    • Exercises about rational and irrational numbers: Epp 4th Edition Section 4.6 # 2 and these Three Extra Questions:
      1. Suppose that \( q=a/b \) is a rational number. What does that tell you about \(a\) and \(b\)?
      2. Suppose that \( q=a/b \) is a rational number and \(q=0\). What does that tell you about \(a\) and \(b\)?
      3. Suppose that \( q=a/b \) is a rational number and \(q \neq 0\). What does that tell you about \(a\) and \(b\)?
    • Exercises to be proven directly, not using contradiction and not proving the contrapositive: Epp 4th Edition Section 4.6 # 4,5,6,7,17
    • Exercises to be proven indirectly, by proving the contrapositive: Epp 4th Edition Section 4.6 # 10,12,14,15,19,21,22,24,25,26,27,28
    • Exercises to be proven indirectly by contradiction: Epp 4th Edition Section 4.6 # 16
    • Remark: Observe that out of all of the exercises that I assign in this section, only one needs to be proven by contradiction!
  • Video: Video for Epp 5th Edition Section 4.7 = Topics from Epp 4th Edition Section 4.6 ( link to video )
  • Notes: Notes from Video for Epp 5th Edition Section 4.7 ( Link to Notes )
  • Handout: Handout on Overuse of the Method of Contradiction ( Link to Handout )
• Class Presentations for Mon Oct 10 Show/Hide Presentations

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  Brittany Poss CP3: The goal is to prove the following Statement A :

  For all \(n\in \mathbb{Z}\), if \(n^2\) is odd then \(n\) is odd.
  1. Write the contrapositive of Statement A .
  2. Prove the contrapositive.

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  Clair Schmidt CP3: The goal is to prove the following Statement B :

  For all real numbers \(a,b,r\) such that \(a,b \in \mathbb{Q}\) and \(b\neq0\), if \(r\) is irrational, then \(a+br\) is irrational.
  1. Write the contrapositive of Statement B .
  2. Prove the contrapositive.

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  Jake Schneider CP3: The goal is to prove the following Statement C : $$\forall a,b,c \in \mathbb{Z} \left( \text{If } a \nmid bc \text{ then } a \nmid b \right)$$ (Note that the symbol \( \nmid\) means does not divide .)

  1. Write the contrapositive of Statement C .
  2. Prove the contrapositive.

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  Hide Mon Oct 10 Presentations

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Wed Oct 12:

• Section to Discuss: Epp 4th Edition Section 4.7: Indirect Argument: Two Classical Theorems
• Exercises: Epp 4th Edition Section 4.7 # 1,2,4,8,11,12,14,15,17,21,22,31
• Video: There is no video prepared for this section of the book.
• Handouts: Handout on Overuse of the Method of Contradiction ( Link to Handout )
• Class Presentations for Wed Oct 12 Show/Hide Presentations

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  Nicole Strang CP3: Find the negation of Statement S : $$\text{Statement }S: \forall n \in \mathbb{Z}, p\in Primes \left(\text{If }p \mid n\text{ then }p \nmid (n+1) \right)$$ (Note that the symbol \( \mid\) means divides and the symbol \( \nmid\) means does not divide .)

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  Hide Wed Oct 12 Presentations

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Take-HomeQuiz Q06: Handed out on Wed Oct 12, Due Fri Oct 14, covering Epp 4th Edition Section 4.6, based on one of the exercises in the list

Fri Oct 14:

• Sections to Discuss:
  • Epp 4th Edition Section 4.7: Indirect Argument: Two Classical Theorems
    • Exercises: Epp 4th Edition Section 4.7 # 1,2,4,8,11,12,14,15,17,21,22,31
    • Video: There is no video prepared for this section of the book.
    • Handouts:
      • Handout on Overuse of the Method of Contradiction ( Link to Handout )
      • Handout on Fermat�s Theorem ( Link to Handout )
  • Epp 4th Edition Section 4.8: The Euclidean Algorithm
    • Exercises: Epp 4th Edition Section 4.8 #13,14,15,16,25,26,27,28,29
    • Video: There is no video prepared for this section of the book.
• Class Presentations for Fri Oct 14 Show/Hide Presentations

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  Rashid Al Busa'idi CP3:

  1. Find the negation of this statement: $$ \forall p\in B \left(\forall x,y,z \in \mathbb{Z}\left(x^p+y^p\neq z^p \right) \right)$$
  2. Find the negation of this statement: $$ \forall n\in C \left(\forall x,y,z \in \mathbb{Z}\left(x^n+y^n\neq z^n \right) \right)$$
  3. Find the contrapositive of this statement: $$\text{If }\left( \forall p\in B \left(\forall x,y,z \in \mathbb{Z}\left(x^p+y^p\neq z^p \right) \right) \right)\text{ then } \left( \forall n\in C \left(\forall x,y,z \in \mathbb{Z}\left(x^n+y^n\neq z^n \right) \right) \right)$$

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  Gretchen Angst CP4: (based on Epp 4th Edition Exercise 4.8#15) The goal is to find the greatest common divisor of \(832\) and \(10933\) two ways

  1. Show how to use Wolfram Alpha. (Project onto the screen.)
  2. Show the steps using the Euclidean algorithm (see Epp 4th Edition pages 222-224)

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  Evan Brooks CP4: (related to Epp 4th Edition Exercise 4.8#15) The goal is to find the greatest common divisor of \(832\) and \(10933\) in a different way, using the prime factorizations .

  1. Show the prime factorizations of \(832\) and \(10933\)
  2. Using those prime factorizations to explain what the greatest common divisor of those two numbers is.

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  Michael Cooney CP4: (related to Epp 4th Edition Exercise 4.8#25) The goal is to find the least common multiple of \(832 \)and \(10933\) using the prime factorizations .

  1. Show the prime factorizations of \(832\) and \(10933\)
  2. Using those prime factorizations to explain what the least common multiple of those two numbers is.

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  Hide Fri Oct 14 Presentations

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Week 9 (Mon Oct 17 through Fri Oct 21)

Mon Oct 17:

• Section to Discuss: Epp 4th Edition Section 4.8: The Euclidean Algorithm
  • Exercises: Epp 4th Edition Section 4.8 #13,14,15,16,25,26,27,28,29
  • Video: There is no video prepared for this section of the book.
• Class Presentations for Mon Oct 17 Show/Hide Presentations

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  Julie Fausnaugh CP4: (Your topic will be part of Monday�s discussion of greatest common divisor and least common multiple , but your topic is background, not related to anything discussed in the book.)
  Prove that if \(j\) and \(k\) are any two integers, then \(max\{j,k\}+min\{j,k\}=j+k\).
  (Hint: Notice that there is an OR statement that can be articulated: $$j \lt k \ \text{ OR } \ j = k \ \text{ OR } \ j \gt k$$ Exploit that to set up a proof by cases.)

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  The next two presentations will be part of the review portion of the class meeting, reviewing concepts from Epp 4th Edition Section 4.4.

  To review the Quotient Remainder Theorem

  • Read Epp 4th Edition Section 4.4 Direct Proof and Counterexample V: Division into Cases and the Quotient Remainder Theorem.
  • Watch the Video for Epp 5th Edition Section 4.5 = Topics from Epp 4th Edition Section 4.4 ( link to video )
  • Read and the accompanying notes ( Link to Notes ).

  Lucas Fernandes CP4: (See note above about reviewing the Quotient Remainder Theorem .) Let \(n\) be an integer. What does the Quotient Remainder Theorem with \(d=4\) say about \(n\)?

  Luke Haskell CP4: (See note above about reviewing the Quotient Remainder Theorem .) Solve Epp 4th Edition exercise 4.4#36: Prove that the product of any four consecutive integers is diviible by 8. (Hint: Let your four consecutive integers be \(n,n+1,n+2,n+3\). Then use the Quotient Remainder Theorem with \(d=4\) to get four possibilities for \(n\).

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  Hide Mon Oct 17Presentations

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Exam X2 on Wed Oct 19 Covering Chapter 4 Show/Hide Exam X2 Info

Fri Oct 21:

• Section to Discuss: Epp 4th Edition Section 5.1 Sequences
• Homework: Epp 4th Edition 5.1#2, 4, 10, 11, 13, 15, 16, 20, 21, 22, 26, 27, 30, 31, 33, 35, 38, 40, 44, 45, 46, 49, 53, 55, 62, 64, 65, 67
• Videos:
  • ( VideoH05.1a ) ( Notes from VideoH05.1a )
  • ( VideoH05.1b ) ( Notes from VideoH05.1b )
• Class Presentations for Fri Oct 19 Show/Hide Presentations

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  Ethan Levingston CP4: (Watch VideoH05.1a , study [Example 2] on page 7 of the notes for VideoH05.1a , and read Epp 4th Edition page 229 Example 5.1.3)

  For the sequence that begins

  $$-1,3,-9,27,-81,...$$
  1. Find an explicit formula of the form \(a_0,a_1,a_2,...\)
  2. Find an explicit formula of the form \(b_1,b_2,b_3...\)

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  Jennifer Lewis CP4: (Watch VideoH05.1a , study [Examples 5,6] on pages 13-14 of the notes for VideoH05.1a , and read Epp 4th Edition page 231 Example 5.1.7)

  For the sum

  $$\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{n\cdot(n+1)}$$
  1. Abbreviate the sum using summation notation .
  2. Find the value of the sum when \(n=1\).

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  Daniel Lipec CP4: (Watch VideoH05.1a , study [Example 4 on pages 12 of the notes for VideoH05.1a , and read Epp 4th Edition page 232 Example 5.1.10)

  For the sum

  $$\sum_{n=1}^{99}\left(\frac{1}{n}-\frac{1}{n+1}\right)$$
  1. Write the sum in expanded form (but still abbreviated using ellipses . That is, use dot dot dot notation, �...�.)
  2. Find the value of the sum.

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  Bradey Lounsbury CP4: (Watch VideoH05.1a and read Epp 4th Edition page 232 Example 5.1.9)

  For the sum $$\sum_{k=1}^{m+1}k^3$$ Rewrite the sum by separating off the final term.

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  Hide Fri Oct 21 Presentations

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Week 10 (Mon Oct 24 through Fri Oct 28)

Mon Oct 24

• Section to Discuss: Epp 4th Edition Section 5.1 Sequences
• Homework: Epp 4th Edition 5.1#2, 4, 10, 11, 13, 15, 16, 20, 21, 22, 26, 27, 30, 31, 33, 35, 38, 40, 44, 45, 46, 49, 53, 55, 62, 64, 65, 67
• Videos:
  • ( VideoH05.1a ) ( Notes from VideoH05.1a )
  • ( VideoH05.1b ) ( Notes from VideoH05.1b )
• Class Presentations for Mon Oct 24 Show/Hide Presentations

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  Bryce Nicholson CP4: (Read Epp Section 5.1, watch VideoH05.1a , and study [Examples 9,10] on page 17 of the notes for VideoH05.1a .)

  Expanded form often looks more intimidating than product notation, but it sometimes makes things more complicated. Consider this telescoping product, presented in expanded form:

  $$\text{Original product }=\left(\frac{1}{1+1}\right)\left(\frac{2}{2+1}\right)\left(\frac{3}{3+1}\right)\dotsm\left(\frac{k}{k+1}\right)$$
  1. Find the value of the product when \(k=2\).
  2. Express the Original Product in product notation.
  3. Express the product from (a) in product notation.

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  Brittany Poss CP4: (Read Epp Section 5.1, watch VideoH05.1a , and study [Examples 9,10] on page 17 of the notes for VideoH05.1a .)

  1. Find the value of this telescoping product.
  $$\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{3}\right)\dotsm\left(1-\frac{1}{1000}\right)$$
  2. Write the product from (a) in product notation.
  3. What do you think would be the value of this telescoping product? Explain.
  $$\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{3}\right)\dotsm\left(1-\frac{1}{n}\right)$$
  4. Write the product from (c) in product notation.

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  Clair Schmidt CP4: (Read Epp Section 5.1, watch VideoH05.1b and study [Example 1] on page 7 of the Notes from VideoH05.1b .)

  1. Show how \(\frac{6!}{4!}\) is found by cancelling factors.
  2. Show how \(\frac{6!}{1!}\) is found by cancelling factors.
  3. Show how \(\frac{6!}{0!}\) is found by cancelling factors.
  4. Show how \(\frac{n!}{(n-3)!}\) is found by cancelling factors (assume that \(3 \leq n\).)
  5. Show how \(\frac{n!}{(n-p+3)!}\) is found by cancelling factors (assume that \(3 \leq p \leq n+3\).)

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  Jake Schneider CP4: (Read Epp Section 5.1, watch VideoH05.1b and study [Example 9] on page 9 of the Notes from VideoH05.1b .)

  1. Show how \(\left( \begin{eqnarray} 7 \\ 4 \end{eqnarray} \right) \) is found by cancelling factors.
  2. Show how \(\left( \begin{eqnarray} 5 \\ 1 \end{eqnarray} \right) \) is found by cancelling factors.
  3. Show how \(\left( \begin{eqnarray} 5 \\ 5 \end{eqnarray} \right) \) is found by cancelling factors.
  4. Show how \(\left( \begin{eqnarray} &n& \\ n&-&1 \end{eqnarray} \right) \) is found by cancelling factors (assume that \(1 \leq n\).)
  5. Show how \(\left( \begin{eqnarray} n+1 \\ n-1 \end{eqnarray} \right) \) is found by cancelling factors (assume that \(1 \leq n\).)

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  Hide Mon Oct 24 Presentations

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Wed Oct 26

• Section to Discuss: Section 5.2 Mathematical Induction I
• Homework: Epp 4th Edition 5.2 # 1, 2, 2, 4, 6, 7, 8, 10, 11, 12, 13, 20, 22, 25, 28
• Video: ( VideoH05.2 ) ( Notes from VideoH05.2 )

Quiz Q07 on Wed Oct 26, covers Section 5.1

Fri Oct 28

• Section to Discuss: Section 5.2 Mathematical Induction I
• Homework: Epp 4th Edition 5.2 # 1, 2, 2, 4, 6, 7, 8, 10, 11, 12, 13, 20, 22, 25, 28
• Video: ( VideoH05.2 ) ( Notes from VideoH05.2 )
• Class Presentations for Fri Oct 28 Show/Hide Presentations

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  Roman Simkins CP3: (To prepare, read Epp 4th Edition Section 5.2, watch VideoH05.2 and read the Notes from VideoH05.2 . Pay particular attention to [Examples 9,10], beginning on page 17 .)

  Suppose that the goal is use the Method of Proof by Induction to prove the following statement: $$\forall n\in \mathbb{Z},n\geq1 \left(1+6+11+16+\dotsm +(5n-4)=\frac{n(5n-3)}{2}\right)$$ Your job is to do the preliminary work :

  1. What is the predicate , \(P(n)\)?
  2. What is the number playing the role of \(a\)?
  3. Write the expression for \(P(a)\).
  4. Write the expression for \(P(k)\).
  5. Write the expression for \(P(k+1)\).
  Note: Don�t do the proof! Just do the preliminary work .

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  Grace Smith CP3: (To prepare, read Epp 4th Edition Section 5.2, watch VideoH05.2 and read the Notes from VideoH05.2 . Pay particular attention to [Examples 9,10], beginning on page 17 .)

  Suppose that the goal is use the Method of Proof by Induction to prove the following statement: $$\forall n\in \mathbb{Z},n\geq1 \left(1^3+2^3+3^3+\dotsm+n^3=\left[\frac{n(n+1)}{2}\right]^2\right)$$ Your job is to do the preliminary work :

  1. What is the predicate , \(P(n)\)?
  2. What is the number playing the role of \(a\)?
  3. Write the expression for \(P(a)\).
  4. Write the expression for \(P(k)\).
  5. Write the expression for \(P(k+1)\).
  Note: Don�t do the proof! Just do the preliminary work .

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  Hide Fri Oct 28 Presentations

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Week 11 (Mon Oct 31 through Fri Nov 4)

Mon Oct 31 Show/Hide topics

Wed Nov 3 Discussed More Examples Involving Induction, had Quiz Q08

Wed Nov 2

• Section to Discuss: Section 5.3 Mathematical Induction II
• Homework: Epp 4th Edition 5.3 # 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 23
• Video: A video has not been prepared for Section 5.3

Quiz Q08 on Wed Nov 4, covers Section 5.2

Fri Nov 4

• Section to Discuss: Section 5.4 Strong Mathematical Induction
• Homework: Epp 4th Edition 5.4 # 1, 4, 5, 9, 10, 13, 17, 18, 23, 24, 29, 30
• Video: A video has not been prepared for Section 5.4
• Handouts: Handout on Induction

Week 12 (Mon Nov 7 through Fri Nov 11)

Exam X3 on Mon Nov 7 Covers Sections 5.1, 5.2, 5.3, 5.4

Wed Nov 9

Sections to Discuss:

• Epp 4th Edition Section 1.2 The Language of Sets (No Video) (#11,12)
• Epp 4th Edition Section 1.3 The Language of Relations and Functions (No Video) (#1,3,5,6,7,9,10,12,13,15,19)
• Epp 4th Edition Section 8.1 Relations on Sets
  • Homework: Epp 4th Edition 8.1 # 1,3,4,7,10,11,12,13,15,16,19,21,22
  • Video: ( video ) ( notes )

Fri Nov 11 is Veterans Day Holiday: No Class

Week 13 (Mon Nov 14 through Fri Nov 18)

Mon Nov 14 Show/Hide topics

Wed Nov 16

• Section to Discuss: Epp 4th Edition Section 8.2 Reflexivity, Symmetry, and Transitivity
• Exercises: Epp 4th Edition 8.2 # 4,5,7,8,9,10,11,13,14,15,16,30,32,33
• Video: ( VideoH08.2 ) ( Notes from VideoH08.2 )

Fri Nov 18

• Section to Discuss: Epp 4th Edition Section 8.3 Equivalence Relations
• Exercises: Epp 4th Edition 8.3 # 3,5,7,15,16a,20,21,28,29,32,42
• Video: ( VideoH08.3 ) ( Notes from VideoH08.3 )

Quiz Q09 on Nov 18 Covers Sections 8.1, 8.2

Week 14 (Mon Nov 21 through Fri Nov 25)

Mon Nov 21 Show/Hide topics

Wed and Fri Nov 23 and 25 are Thanksgiving Break

Week 15 (Mon Nov 30 through Fri Dec 4)

Mon Nov 28 Show/Hide topics

Wed Nov 30 Show/Hide topics

Quiz Q10 on Wed Nov 30 Covers Section 8.4

Fri Dec 2 Show/Hide topics

Week 16 (Mon Dec 7 through Fri Dec 11)

Final Exam on Mon Dec 5 from 10:10am - 12:10pm in Morton 218 Show/Hide Final Exam Info