Math 235 - Foundations of Mathematical Proof
-- Spring 2024 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.
Listings go in order of most RECENT date on top.
*** WEEK TWELVE IS BELOW. ***
- From Wednesday, Apr. 10, 2024 - Proof by Mathematics Induction, Part III
- Study your notes.
- Review inequality algebra as needed. (Use Google judiciously.)
- Read/reread Section 2.4 as needed.
- Beware imitating the book's write-ups; they are a bit sloppy in terms of rigor AND flow!
- HW #10 is due Friday, Apr. 12, by 4pm.
- Confirmed From Monday, Apr. 8 - Proof by Mathematical Induction:Part II
- Study your notes.
- 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 #10 is due Friday, Apr. 12, by 4pm.
*** WEEK ELEVEN IS BELOW. ***
- Confirmed From Friday, Apr. 5, 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 plus 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 #10 is due Friday, Apr. 12, by 4pm.
- Confirmed From Wednesday, Apr. 3, 2024 - Functions: One-to-one Proofs
- Study your notes.
- No new reading, but I will look for a few more examples beyond our own textbook
- Review the three Short Practice tasks we discussed from #15(a),(b),(d) on p.229.
- Review notes and/or text from Discrete Math.
- HW #9 is due by 4pm Friday, Apr. 5.
- Confirmed From Monday, Apr. 1, 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 #9 is due by 4pm Friday, Apr. 5.
*** WEEK TEN IS BELOW. ***
- Confirmed From Friday, Mar. 29, 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 by 4pm Friday, Apr. 5.
- Confirmed From Wednesday, Mar. 27, 2024 - Functions: Basic (Set) Proofs
- Study your notes.
- Read/reread Section 4.1 p.203 through end of Figure 4.1.3 on p.206.
- Review notes and/or text from Discrete Math.
- If you want to re-earn points on HW #6 and #7, please present to me this week.
- HW #8 is due by 4pm this Friday, Mar. 29.
- Confirmed From Monday, Mar. 25, 2024 - Functions: Review and Basic (Set) Proofs
- Study your notes.
- Read Section 4.1 p.203 through end of Figure 4.1.3 on p.206.
- Review notes and/or text from Discrete Math.
- If you want to re-earn points on HW #6 and #7, please present to me this week.
- HW #8 is due by 4pm this Friday, Mar. 29.
*** WEEK NINE IS BELOW. ***
- Confirmed From Friday, Mar. 22, 2024 - Exam #2
- Exam #2 is in class today.
- Your individualized take-home problem is due by 4pm Monday, Mar. 25.
- HW #8 is due NEXT Friday, Mar. 29.
- Confirmed From Wednesday, Mar. 20, 2024 - Relations: Review
- Study your notes.
- Read/reread:
- Section 3.1: pp.153-160
- You may be confused by three of the Example proofs in Section 3.1: they do NOT fit all our requirements and practices for this course (or in general, in one case).
- So also read Summary #1: Corrections to Proofs about Relations in Book.
- The Summary rewrites the flawed proofs to meet our standards.
- This should help you more reliably use them for guidance on your own proofs.
- Section 3.2: pp.163-166
- The definition on p.166 is not in notes, but also not hard.
- (iii) in the proof on p.166 takes "even+even=even" as an axiom.
- Notes and/or text from Discrete Math
- Exam #2 is in class on FRIDAY, Mar. 22.
- It will have a combined existence/uniqueness take-home problem due Monday, Mar. 25.
- HW #8 is due NEXT Friday, Mar. 29.
- Confirmed From Monday, Mar. 18, 2024 - Relations: R, S, T Proofs
- Study your notes.
- 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."
- Review notes and/or text from Discrete Math.
- Exam #2 is in class on FRIDAY, Mar. 22.
- There is no HW due until the week following Exam #2.
- You can earn back missed points on HW #6 and #7 by meeting in my office to discuss/present corrected versions.
- Prepare to discuss #20 on p.103 as Short Practice on Wednesday.
- None earn an A - be able to describe what's wrong.
*** Spring Break IS BELOW. ***
- From Friday, Mar. 15, 2024 - Spring Break
- From Wednesday, Mar. 13, 2024 - Spring Break
- From Monday, Mar. 11, 2024 - Spring Break
*** WEEK EIGHT IS BELOW. ***
- Confirmed From Friday, Mar. 8, 2024 - Relations: Review and Basic (Set) Proofs
- Study your notes.
- Read Section 3.1: pp.153-160.
- Beware 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.
- Exam #2 is in class on FRIDAY, Mar. 22.
- There is no HW due until the week following Exam #2.
- HW that I have now will come back to you ASAP.
- There will be bonus immediately on HW #5.
- There will also be the opportunity to earn back missed points on HW #6 and #7 by meeting in my office to discuss.
- Confirmed From Wednesday, Mar. 6, 2024 - Sets: General Proofs
- Study your notes.
- Read/reread:
- Review definitions and outlines carefully.
- 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 4pm on Friday, Mar. 8.
- Confirmed From Monday, Mar. 4, 2024 - Sets: Identities; General Proofs
- Study your notes.
- Read/reread:
- Identity proof reading listed from Friday, Mar. 1
- What we're beginning now simply merges ALL of the proof styles we've covered since the start of the course. Review as you need.
- Here are some good examples that are less "definition-chase" than recent subset/not/= proofs or chain-style identity proofs:
- 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
- Here is a nice Guide to Set Theory Proofs from Stanford University that gives one more perspective on subset/not/= proofs and general set proofs.
- HW #7 is due by 4pm on Friday, Mar. 8.
*** WEEK SEVEN IS BELOW. ***
- Confirmed From Friday, Mar. 1, 2024 - Sets: Identity Proofs
- Study your notes.
- Read 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!
- Also read p.47's top Example and Proof for another example of Chain-Style Proof that isn't using sets.
- HW #7 is due by 4pm on Friday, Mar. 8.
- Confirmed From Wednesday, Feb. 28, 2024 - Sets: Equality/Not Proofs; review of operations
- Study your notes.
- Review your notes and work from Discrete Math.
- Read/reread:
- 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 p.95 and top Example on p.96
- Section 2.2 Definition on p.98 down TO Theorem 2.2.2 (omit that Theorem)
- top half of p.100, down TO Theorem 2.2.3
- HW #6 is due by 4pm on Friday, Mar. 1.
- Prepare to discuss pp.94-95 #19 as Short Practice on Friday. (I will only pick a few, not all.)
- Confirmed From Monday, Feb. 26, 2024 - Sets: Brief review; 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:
- p.85 through end of top proof on p.88
- p.90 everything below Figure 2.1.1
- Theorem 2.1.5 and its proof on p.91
- HW #6 is due by 4pm Friday, Mar. 1.
- Prepare to discuss pp.94-95 #19 as Short Practice in class briefly on Friday. (I will only pick a few parts, not all.)
*** WEEK SIX IS BELOW. ***
- Confirmed From Friday, Feb. 23, 2024 - Uniqueness proofs
- Study your notes.
- Read:
- Section 1.6 mid p.58 through mid p.59
- Section 1.7 "(E!x)P(x)" on pp.71-72
- The text discusses existence-uniqueness proofs here, not just uniqueness.
- HW #5 is due by 4pm this Friday, Feb. 23.
- Confirmed From Wednesday, Feb. 21, 2024 - Mixed quantifier proofs: Catch-up, analyze
- Study your notes.
- There is no additional supporting reading.
- HW #5 is due by 4pm this Friday, Feb. 23.
- Confirmed From Monday, Feb. 19, 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
- Your Exam #1 take-home problem is due by 4pm today.
- HW #5 is due by 4pm this Friday, Feb. 23.
- We WILL do a Short Practice discussion in class on Wednesday, but you are welcome to PASS if you don't have time/feel comfortable.
- These are the problems to analyze: #9(a), (c), (d), (e), (h), (j) from pp.62-63.
*** WEEK FIVE IS BELOW. ***
- Confirmed From Friday, Feb. 16, 2024 - Exam #1
- Exam #1 is in class today.
- Your individualized take-home problem is due by 4pm on Monday, Feb. 19.
- HW #5 is due by 4pm on Friday, Feb. 23.
- Confirmed From Wednesday, Feb. 14, 2024 - Other Proof Styles for "For all," "There exists"
- Study your notes.
- Read/reread Section 1.6 pp.51 through end of TOP Proof on p.57.
- 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.
- Exam #1 is in class Friday, Feb. 16.
- The link opens the PDF Topics List/Objectives List.
- There is no HW due until after the exam. (I'll give you HW #5 as you leave the exam.)
- Confirmed From Monday, Feb. 12, 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.
- Exam #1 is in class Friday, Feb. 16.
- The link opens the PDF Topics List/Objectives List.
- There is no HW due until after the exam.
- In class on Wednesday, we will review with a little Short Practice: p.63 #9fg, p.75 #11cdg, p.83 #21cg.
- Come prepared to discuss, including which proof style the writer is trying.
*** WEEK FOUR IS BELOW. ***
- Confirmed From Friday, Feb. 9, 2024 - Direct Proof of ``For all''
- Study your notes.
- Read/ Section 1.6 pp.51 through end of top paragraph on p.53.
- Exam #1 is in class Friday, Feb. 16.
- The link opens the PDF Topics List/Objectives List.
- There is no HW due until after the exam.
- Confirmed From Wednesday, Feb. 7, 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, Feb. 9, by 4pm.
- Exam #1 IS in class next Friday, Feb. 16.
- Confirmed From Monday, Feb. 5, 2024 - Biconditionals: Two-Part Proof
- Study your notes.
- 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, Feb. 9, by 4pm.
***** WEEK THREE IS BELOW. *****
- Confirmed From Friday, Feb. 2, 2024 - Proof by XX; Using lemmas
- Study your notes.
- Read/reread:
- Section 1.5: p.43 through end of Proof on p.45
- Section 1.5: Example on bottom of p.47 through end of Section (on 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, Feb. 2, by 4pm.
- HW #4 is due by 4pm next Friday, Feb. 9.
- You are equipped to tackle Problems #1 and #2.
- For Short Practice 3 discussion on Monday, Feb. 5, give grades of A or "not an A" to the attempted proofs in #12 on pp.50-51. Prepare to support your claims.
- Confirmed From Wednesday, Jan. 31, 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
- Section 1.5: BOTTOM Example on p.47 through end of Section (on 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 FRIDAY, Feb. 2, by 4pm.
- Confirmed From Monday, Jan. 29, 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!
- HW #3 is due FRIDAY, Feb. 2, by 4pm.
- The PDF *includes* Part A that you received on Friday.
***** WEEK TWO IS BELOW. *****
- Confirmed From Friday, Jan. 26, 2024 - "Or Conclusion"-style Proof
- HW #2 is due TODAY by 4pm.
- Study your notes.
- Read Section 1.7 "If P, then Q1 v Q2" (and example) on p.70.
- BEWARE: the authors use x,y as both logic variables AND function variables.
- That's NOT a good idea at our level of study.
- HW #3 is due FRIDAY, Feb. 2, by 4pm.
- Confirmed From Wednesday, Jan. 24, 2024 - Proof by Cases, continued
- Study your notes.
- Read/reread (all items come from our text unless otherwise indicated):
- Preface to the Student: pp.xii-xiv
- Section 1.7: p.67 through top paragrapn on p.68
- Xeroxed handout "Some Mathematical Conventions" (also posted in D2L reading Content)
- pp.64-66
- mid p.36 ("When either P or Q...") through 1st proof on p.38
- 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
- Xeroxed supplement from Fraleigh (posted in D2L Content)
- HW #2 is due FRIDAY, Jan. 26, by 4pm.
- Confirmed From Monday, Jan. 22, 2024 - Proof by Cases
- Study your notes.
- Read by FRIDAY (all items come from our text unless otherwise indicated):
- Preface to the Student: pp.xii-xiv
- Section 1.7: p.67 through top paragrapn on p.68
- Xeroxed handout "Some Mathematical Conventions" (also posted in D2L reading Content)
- pp.64-66
- By Wednesday, read:
- Section 1.4: Below second blue box on p.33 through end of 1st proof on p.38.
- ESPECIALLY NOTE mid p.36 ("When either P or Q...") through 1st proof on p.38
- 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.
- HW #2 is due FRIDAY, Jan. 26, by 4pm.
- With just a 5-day turn-around, this one is shorter than what you should expect later in the course.
***** WEEK ONE IS BELOW. *****
- Confirmed From Friday, Jan. 19, 2024 - What is proof?, Direct Proof
- Study your notes.
- Read:
- 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.
- Xeroxed excerpt from Epp (scanned as PDF in D2L reading Content)
- Reminder: HW #1 is due MONDAY, Jan. 22, by 4pm.
- I'll distribute HW #2 in class Monday; it's due Friday so will be a little shorter than typical.
- For Monday's class, attempt to prove the following, as a Short Practice (we'll discuss it in some way, NOT turn it in.): Let L belong to Z. If L+6 is even, then L^2-8 is even.
- Confirmed From Wednesday, Jan. 17, 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 MONDAY, Jan. 22, by 4pm.
Dr. J. Lyn Miller, Dept. of Mathematics and Statistics, Slippery Rock University