Math 235 - Foundations of Mathematical Proof
Fall 2024 Daily Assignments and Tasks
- My teaching style is very adaptive, so I only post class assignments and tasks as we become ready to cover that material.
- These are labeled Preliminary before class.
- AFTER class, I adjust the list to accommodate for any flexing we did DURING class that day, and mark the listing Confirmed.
-
Listings go in order of most RECENT date on top.
*** WEEK THIRTEEN IS BELOW. ***
- CONFIRMED From Monday, Nov. 18, 2024 - Proof by Mathematical Induction:Part II
- Study your notes and the second induction activity.
- Read/reread Section 2.4 (pp.114-123).
- Beware imitating the book's write-ups; they are a bit sloppy in terms of rigor AND flow!
- HW #11 is due by 5pm TODAY.
- Exam #3 is Monday, Nov. 25 (one week from today). (Link opens Topics List)
***** WEEK TWELVE IS BELOW. *****
- Preliminary From Friday, Nov. 15, 2024 - Proof by Mathematical Induction: Part I
- Study your notes and the Activity: A Template for Proof by Induction
- Read Section 2.4 pp.114-117 and Example using PI-notation plus its Proof on bottom p.118 through p.119.
- Beware imitating the book's write-ups; they are a bit sloppy in terms of rigor AND flow!
- HW #11 is due Monday, Nov. 18, by 5pm.
- CONFIRMED From Wednesday, Nov. 13, 2024 - Functions: One-to-one Proofs, Review
- Study your notes and the Onto/One-to-One Activity.
- Solutions were distributed in class and will also be posted in D2L Content tomorrow.
- Read/reread selections from Monday and earlier.
- Review notes and/or text from Discrete Math.
- HW #10 is due Friday, Nov. 15, by 4pm.
- HW #11 is due by 5pm MONDAY, Nov. 18in class or my box.
- Confirmed From Monday, Nov. 11, 2024 - Functions: One-to-one Proofs
- Study your notes.
- Read Section 4.3: defn on p.224 through end of p.225.
- Review notes and/or text from Discrete Math.
- HW #10 is due Friday, Nov. 15, by 4pm.
- HW #11 is due by 5pm MONDAY, Nov. 18in class or my box.
***** WEEK ELEVEN IS BELOW. *****
- CONFIRMED From Friday, Nov. 8, 2024 - Functions: Onto Proofs
- Study your notes.
- Read Section 4.3 p.222 through end of Theorem 4.3.2 proof on p.224.
- Review notes and/or text from Discrete Math.
- HW #9 is due in D2L Quizzes by 11:59pm TODAY.
- HW #10 is due NEXT Friday, Nov. 15, by 4pm in my box/office/class and consists of three parts:
- (1) The back page from our function review Activity last class
- (2) Re-Dos of the three proof launches from Exam #2
- (3) A stand-alone list of ALL formal definitions/statements we've covered (see the names on Topics Lists #1 and #2) and ALL proof-style outlines.
- HW #11 on function proofs will be distributed next class and be due on MONDAY, Nov. 18, by 5pm.
- CONFIRMED From Wednesday, Nov. 6, 2024 - Review, Revisions
- Study your returned Exam #2 In-Class portion and relevant notes.
- HW #9 is due in D2L Quizzes by 11:59pm on Friday, Nov. 8.
- Complete rewrites of the TWO problems circled in green on YOUR individual Exam #2 are due Friday, Nov. 8, by 4pm if possible.
- This is a short turn-around, so let me know if you need an extension.
- HW #10 is due NEXT Friday, Nov. 15, by 4pm in my box/office/class and consists of three parts:
- (1) The back page from our function review Activity last class
- (2) Re-Dos of the three proof launches from Exam #2
- (3) A stand-alone list of ALL formal definitions/statements we've covered (see the names on Topics Lists #1 and #2) and ALL proof-style outlines.
- CONFIRMED From Monday, Nov. 4, 2024 - Functions: Definition Review
- Study your notes and the review Activity
- The front side of the Activity counts as the DP #11 score.
- The BACK side is becoming part of HW #10.
- Read:
- Section 4.1 p.203 through end of Figure 4.1.3 on p.206
- Section 4.3 pp.222 up TO Theorem 4.3.3 on p.225
- Review notes and/or text from Discrete Math.
- HW #8 is due by 5pm TODAY in class or my box.
- HW #9 is the Function Definition "quiz" in D2L Quizzes.
- You have unlimited attempts on it until 11:59pm Friday, Nov. 8.
- You may also use static resources and also work together if you like.
- Some phrasings or notation may be things you have to look up from Discrete Math.
***** WEEK TEN IS BELOW. *****
- CONFIRMED From Friday, Nov. 1, 2024 - Relations: R,S,T Proofs, continued
- Study Activity: More Practice with Relation Proofs
- Solutions were distributed in class and are also posted in D2L Content.
- Working on the Activity counts as 3/3 for DP 9 and DP 10.
- Read/reread supporting list from lastWednesday.
- The online portion of Exam #2 is in D2L Quizzes and due by 11:59pm TONIGHT.
- HW #8 is due MONDAY, Nov. 4 by 5pm.
- CONFIRMED For Wednesday, Oct. 30, 2024 - Exam #2
- Exam #2 is in class today.
- The randomized Take-Home portion is on D2L, in the Quizzes area.
- The Take-home portion is due by 11:59pm on Friday, Nov. 1.
- Instructions about what you can and cannot do in taking the Take-Home are in D2L. Honesty and integrity are expected.
- HW #8, is due MONDAY, Nov. 4, by 5pm.
- CONFIRMED From Monday, Oct. 28, 2024 - Relations: R,S,T Proofs, continued
- Study your notes.
- Read/reread supporting list from lastWednesday.
- Exam #2 is NEXT class: Wednesday, Oct. 30 for the in-class portion.
- The online portion will be due in D2L Quizzes by 11:59pm Friday, Nov. 1.
- It will open no later than the end of the in-class portion.
- HW #8 is due MONDAY, Nov. 4 by 5pm.
***** WEEK NINE IS BELOW. *****
- CONFIRMED From Friday, Oct. 25, 2024 - Relations: R,S,T Proofs, continued
- Study your notes.
- Read/reread supporting list from Wednesday.
- Exam #2 has moved to NEXT Wednesday, Oct. 30 for the in-class portion.
- The online portion will be due in D2L Quizzes by 11:59pm Friday, Nov. 1.
- There is no HW due this Friday because of the upcoming exam.
- HW #8 is due MONDAY, Nov. 4.
- CONFIRMED From Wednesday, Oct. 23, 2024 - Relations: R, S, T Proofs
- Study your notes.
- Read:
- Section 3.1: pp.153-160 as Discrete Math review
- Do NOT imitate the 4th sentence on this page ("Clearly, RoS...")!
- Just because two formulas don't LOOK the same doesn't mean they aren't equal (consider sin^2 x + cos^2 x, which looks nothing like the number 1 visually).
- Also use actual SENTENCES in proofs, not the logic symbols shown in the Theorem 3.1.2(b) proof. Those are BAD news!
- Review notes and/or text from Discrete Math.
- Read Section 3.2: pp.163-166.
- All proofs in the book are for "yes" answers to "Is it refl/symm/trans?"
- Study your notes for counterexamples that prove "no."
- Exam #2 has moved to NEXT Wednesday, Oct. 30 for the in-class portion.
- The online portion will be due in D2L Quizzes by 11:59pm Friday, Nov. 1.
- There is no HW due this Friday because of the upcoming exam.
- HW #8 is due MONDAY, Nov. 4.
- CONFIRMED From Monday, Oct. 21, 2024 - Sets: Wrap-Up
- Study your notes, Discussion/Participation #7, and Discussion/Participation #8.
- Typed solutions will be posted in D2L Content when ready.
- Read/reread already assigned support and handout "Examples of General Set Proofs" in D2L Content.
- Exam #2 is in class on MONDAY, Oct. 28.
- HW #7 is due by 5pm today.
- There is no HW due this Friday because of Monday's exam.
***** WEEK EIGHT IS BELOW. *****
- Confirmed From Friday, Oct. 18, 2024 - Sets: General Proofs
- Study your notes.
- Read/reread:
- Review definitions and outlines carefully from the entire course.
- Examples from textbook:
- Section 2.1: p.89 Theorem 2.1.3 and proof
- Section 2.1: p.91 Theorem 2.1.5 and proof
- Section 2.2: pp.96-97 Theorem 2.2.1(p) and proof
- Section 2.2: pp.98-99 Theorem 2.2.2(e) and proof
- Guide to Set Theory Proofs from Stanford University
- HW #7 is due by 5pm on Monday, Oct. 21.
- CONFIRMED From Wednesday, Oct. 16, 2024 - Sets: Identity Proofs
- Study your notes, and review your notes/work from Discrete Math.
- Read:
- Section 2.1 pp.88-91
- Warning In the top proof on p.89, Case 1 should begin "ASSUME x>=1," NOT "if x>=1."
- Section 2.2 pp.95-101.
- Do NOT imitate the book's "grammar parentheses" as in 2.2.1(m) and 2.2.3(a)! Use commas or semicolons instead.
- Also remember that rules for logic are mostly in Theorem 1.1.1 on p.5.
- Do NOT imitate the book's "if" phrasing in 2.2.3(e). Use "Assume" instead!
- HW #7 is due by 5pm MONDAY, Oct. 21.
- CONFIRMED From Monday, Oct. 14, 2024 - Sets: Subset/Not-a-Subset Proofs
- Study your notes.
- Review your notes and work from Discrete Math.
- Read:
- Xeroxed excerpt from Houston (in D2L reading Content)
- Appendix I pp.379-380
- Section 2.1:
- pp.85-88
- p.90 everything below Figure 2.1.1
- Theorem 2.1.5 and its proof on p.91
- HW #6 is due by 5pm TODAY.
- HW #7 is due by 5pm NEXT Monday, Oct. 21.
***** WEEK SEVEN IS BELOW. *****
- CONFIRMED From Friday, Oct. 11, 2024 - Presentation Prep
- All SRU classes have moved to remoted instruction today due to a power outage.
- Spend our class time studying your graded HW #5 and answer key (in D2L Content), to prepare for your presentation "Re-Do."
- See email and Wednesday notes for a reminder of the guidelines.
- HW #6 is now due on MONDAY by 4pm.
- CONFIRMED From Wednesday, Oct. 9, 2024 - Review and regroup: Proof Styles Based on Logic Structure
- Study your notes and your returned HW #5.
- Take note of items in our Learning Skills list that may help you going forward.
- There is no additional reading today.
- HW #6 is due by 4pm next Friday, Oct. 11.
- An ORAL PRESENTATION Re-Do of parts of HW #5 must be finished by 4pm Friday, Oct. 18.
- I'll restore your lost points on your lowest "for all" proof and your lowest "there exists" proof, after you:
- Present a correct version of your lowest graded "for all" proof, and
- Present MY choice of ANY "there exists" proof, graded or not
- You must present at the board WITHOUT referring to notes/HW.
- Set up a time to meet with me, or let me know which Student Hour time you'll be coming, so we aren't all crowded in at once right at the end.
- From Monday, Oct. 7, 2024 - Fall Break
- There are no SRU classes today.
***** WEEK SIX IS BELOW. *****
- CONFIRMED From Friday, Oct. 4, 2024 - Other Proofs Styles Mixed with Quantifier Proofs
- Note: D2L access to our etext is not working for me. I cannot confirm the reading below until I am in the office sometime over break to check my physical copy of the book. All else in this listing is accurate.
- Study your notes.
- Read/reread Section 1.6 pp.51 through end of SECOND Proof on p.58.
- This supporting reading shows MORE examples of our earlier proof styles now mixed with quantifiers.
- In the p.54 proof, there MUST be words beginning each of those lines of computation.
- Those lines are NOT just one big long, single simplification.
- Also recall there is a typo in the proof of Theorem 1.6.2: The fractions (x+y)/z should have the number 2 as a denominator!
- HW #6 is due by 4pm next Friday, Oct. 11.
- CONFIRMED From Wednesday, Oct. 2, 2024 - Mixed quantifier proofs
- Study your notes.
- Read/reread:
- Section 1.6: Theorem 1.6.2 on p.57 through end of SECOND Proof on p.58
- WARNING - there is a typo in the proof of Theorem 1.6.2.
- The fractions (x+y)/z should have the number 2 as a denominator!
- Section 1.6: starting with "Proving that..." on p.60 to end of section
- Section 1.7: "(forall x)P(x)" subsection at bottom of p.70
- Section 1.7: "(exists x)P(x)" subsection at top of p.71
- HW #5 is due by 4pm this Friday, Oct. 4.
- CONFIRMED From Monday, Sept. 30, 2024 - Constructive Proof of ``There exists''
- Study your notes.
- Read/ Section 1.6 mid p.54 (just above Outline box) through end of top Proof on p.55.
- HW #5 is due Friday, Oct. 4, by 4pm.
***** WEEK FIVE IS BELOW. *****
- CONFIRMED For Friday, Sept. 27, 2024 - Exam #1
- Exam #1 is in class today.
- You'll receive your randomized Take-Home Problem as you leave.
- HW #5, Part A is available now. HW #5 is due next Friday, Oct. 4, by 4pm.
- CONFIRMED For Wednesday, Sept. 25, 2024 - HW Feedback and Review
- Study your graded HWs from the course so far plus Spring 2023 Exam #1.
- Exam #1 is NEXT class, Friday, Sept. 27. (Link opens Topics List)
- You'll receive your randomized Take-Home Problem as you leave the exam.
- HW #5 handouts will also be available at the exam on Friday.
- CONFIRMED For Monday, Sept. 23, 2024 - Direct Proof of "For all"
- Study your notes.
- Read/:
- "Universal Conditional" excerpt from Epp (distributed in class, aslo in D2L Content)
- Section 1.6 pp.51 through end of top paragraph on p.53
- Exam #1 is in class Friday, Sept. 27.
- The link opens the PDF Topics List/Objectives List.
- There is no HW due until after the exam.
***** WEEK FOUR IS BELOW. *****
- CONFIRMED From Friday, Sept. 20, 2024 - Proof Analysis and Launches
- Study your notes, Discussion/Participation #4, and Activity: Proof Launches.
- Solutions to DP #4 will be uploaded to D2L Content over the weekend.
- We will discuss the Activity on Monday, and solutions will then be uploaded to D2L as well.
- There is no new reading.
- HW #4 is due by 4pm today.
- HW #5 will not be due until the week after Exam #1. (I will distribute it at the exam.)
- Exam #1 is next Friday, Sept. 27. (Link opens Topics List)
- CONFIRMED From Wednesday, Sept. 18, 2024 - Biconditionals: Chain-style Proof
- Study your notes.
- Read Section 1.5 bottom of p.46 (immediately under the end of the Proof) and top Example and Proof on p.47.
- HW #4 is due FRIDAY, Sept. 20, by 4pm.
- CONFIRMED For Monday, Sept. 16, 2024 - Biconditionals: Two-Part Proof
- Study your notes and the in-class Activity: Group Biconditional Proofs.
- Read:
- Section 1.5: Beginning with "Proofs of biconditional sentences often..." near bottom p.45 through end of Proof on p.46
- The Proof's (i) is the => direction, while (ii) is the <= direction.
- Label your own proofs with the arrow notation.
- Section 1.7: "P if and only if Q" on p.70
- HW #4 is due FRIDAY, Sept. 20, by 4pm. (Refresh the page to see the full handout, not just Part A.)
***** WEEK THREE IS BELOW. *****
- CONFIRMED From Friday, Sept. 13, 2024 - Proof by CONTRADICTION (XX)
- Study your notes.
- Read/reread:
- Section 1.5: p.43
- Section 1.5: p.45 ONLY the Example and Proof in the top half of the page
- Section 1.5: BOTTOM Example on p.47 through the Proof by Contradiction at the top of p.48
- Section 1.7: "Any statement - Try a proof by contradiction" bottom p.69 and top 3 lines on p.70
- HW #3 is due TODAY by 4pm.
- HW #4 is due FRIDAY, Sept. 20, by 4pm.
- CONFIRMED From Wednesday, Sept. 11, 2024 - Proof by Contrapositive
- Study your notes.
- Read:
- Section 1.5: p.42 through top 4 lines on p.43
- Section 1.5: p.47 BOTTOM Example, Direct Proof, Proof by Ctp
- Section 1.7: "If P, then Q - Try a proof by contraposition" from bottom p.68 through end of example and proof atop p.69
- Notice the CORRECT distinction between function variable x and proof variable t here!
- They do use some "previous results" in the sentence "Because t^2+t is even..." - Beware! I would require the DEFINITION of even for us!
- HW #3 is due FRIDAY, Sept. 13, by 4pm.
- CONFIRMED From Monday, Sept. 9, 2024 - Proof by Cases, Proof by "Or Conclusion" style
- Study your notes.
- Read/reread:
- Supporting reading from Friday, Aug. 30-Friday, Sept 6
- About "without loss of generality":
- Section 1.4: p.38, starting with "Finally, there are proofs by exhaustion..."
- Section 1.7: p.69
"'If P, then Q' - Try a proof by cases" paragraph, example, and proof
- About "or conclusion style":
- Section 1.4: p.36 paragraph near bottom that begins "To prove a conditional sentence whose consequent is a disjunction..."
- Section 1.7: p.70 "If P, then Q1 v Q2" paragraph, example, and proof
- BEWARE: the authors use x,y as both proof variables AND function variables here.
- That's NOT a good idea at our level of study!
- HW #3 is due FRIDAY, Sept. 13, by 4pm.
***** WEEK TWO IS BELOW. *****
- CONFIRMED For Friday, Sept. 6, 2024 - Proof by Cases
- Study your notes.
- Read/reread:
- Section 1.4: mid p.36 ("When either P or Q...") through end of section
- CAUTION: DO NOT USE "if" intros for cases, as the proof on p.37 does.
- INSTEAD, use "suppose" language like the 1st proof on p. 38.
- Section 1.8: p.76 only, and contrast it to ...
- Xeroxed supplement from Fraleigh (posted in D2L Content)
- HW #3 is due FRIDAY, Sept. 13, by 4pm.
- For MONDAY: Prepare to briefly discuss Problem #11, p.41, in class.
- CONFIRMED For Wednesday, Sept. 4, 2024 - Direct Proof
- Study your notes.
- Read/reread supporting reading from Friday, Aug. 30.
- Text: Section 1.4: pp.34- end of SECOND proof on p.36
- HW #2, Part A - DUE Friday, Sept. 6, by 4pm; OMIT Problem #1(e).
- Remember always to leave lots of room for my written feedback .
- You may put your physical copy in my Department mailbox, or bring it to my office.
- In unusual circumstances, I can accept a PDF-format as ONE file sent via email, but I prefer the physical original.
- You are also welcome to give me your physical HW at class-time.
- Monday, Sept. 2, 2024 - Labor Day
- There are no SRU classes today due to the Federal holiday.
***** WEEK ONE IS BELOW. *****
- Confirmed From Friday, Aug. 30, 2024 - What is proof?, Direct Proof
- Study your notes.
- Read:
- Text: Preface to the Student: pp.xii-xiv
- Text: Section 1.4: pp.28-33
- Omit the 2nd example on p.31.
- Beware typo: In the 1st example on p.31, we should see x->2, NOT x->0.
- Text: pp.64-66
- Text: Section 1.7: p.67 through top paragraph on p.68
- Xeroxed excerpt from Epp (scanned as PDF in D2L reading Content)
- Xeroxed handout "Some Mathematical Conventions" (also posted in D2L reading Content)
- HW #2, Part A - DUE DATE TO BE DECIDED NEXT CLASS - for now, do attempt #1-2
- Remember always to leave lots of room for my written feedback .
- You may put your physical copy in my Department mailbox, or bring it to my office.
- In unusual circumstances, I can accept a PDF-format as ONE file sent via email, but I prefer the physical original.
- You are also welcome to give me your physical HW at class-time.
- Confirmed For Wednesday, Aug. 28, 2024 - Review of Logic
(continued)
- Study your notes.
- Review as needed from:
- Your relevant notes and text from Discrete Math
- I've put some excerpts from last semester's text in D2L Content.
- Sections 1.1-1.3 in our text (pp.1-27)
- ESPECIALLY the middle of p.14 in our text
- HW #1 is due FRIDAY, Aug. 30, by 4pm.
- Remember always to leave lots of room for my written feedback .
- You may put your physical copy in my Department mailbox, or bring it to my office.
- In unusual circumstances, I can accept a PDF-format as ONE file sent via email, but I prefer the physical original.
- You are also welcome to give me your physical HW at class-time.
- Confirmed For Monday, Aug. 26, 2024 - Course Intro, Review of Logic
- Study your notes.
- Read the syllabus.
- Review as needed from:
- Your relevant notes and text from Discrete Math
- I've put some excerpts from last semester's text in D2L Content.
- Sections 1.1-1.3 in our text (pp.1-27)
- ESPECIALLY the middle of p.14 in our text
- HW #1 is due FRIDAY, Aug. 30, by 4pm.
- Remember always to leave lots of room for my written feedback .
- You may put your physical copy in my Department mailbox, or bring it to my office.
- In unusual circumstances, I can accept a PDF-format as ONE file sent via email, but I prefer the physical original.
- You are also welcome to give me your physical HW at class-time.
Dr. J. Lyn Miller, Mathematics and Statistics, Slippery Rock University