Foundational Discrete Mathematics

Announcement

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3/20/2025 – The Practice Exam for Unit 3 is now posted below and is due 4/8/2025, as it says on the syllabus. Also, I will be handing back the unit 2 exams on 3/25/2025. On most of the exams I have listed the questions you need to redo in order to earn back upto 40% of the points you lost. For these you need to:
1. Redo the indicated problems in full making sure they are written up neatly in full sentences.
2. Write an explanation of what you did wrong when answering the question on the exam.
3. Turn in the redos, the explanations of what you did wrong, and your original exam on 4/1/2025.
Finally, note that we will not have class on Friday 5/2/2025. The content for that class will be split between the classes before and after that date.
3/4/2025 – There were enough errors on the exam that rather than picking out specific problems from your exams, I put together a common redo exam here: Exam 1 Redo Due 3/11/2025. I will give you all a paper copy of this when I hand back the exam on Tuesday. The Unit 2 Exam practice exam is also now posted below.

Class Calendar

There is a proposed calendar on the syllabus, but here I will record what we actually get through in each class.

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4/15/2025 – We looked at more of the Combinatorics notes, we made it as far as introducing factorials, \(n!\), permutations, \(_nP_k\), and combinations, \(_nC_k=\binom{n}{k}\).
4/11/2025 – Exam on Sets, Relations, and Functions
4/8/2025 – We started going over the Combinatorics Handout after answering questions about the practice exam. The exam is on Friday.
4/4/2025 – Finished Function Material
4/1/2025 – We looked at equivalence class and then started looking at functions. We will finish the material on Functions on 4/4/2025. Remember that the practice exam is due on 4/8/2025.
3/28/2025 – We finished most of the material on relations, we will look at equivalence classes next class and then move onto functions.
3/25/2025 – We continued to work through the packet on Sets, Relations, and Functions. You also got back the second exam. Don’t forget that redos are due on Tuesday April 1st, they must be done neatly in complete sentences, and you must include the redone problems, explanations of what you did wrong on those problems, and turn in the original exam with the redos.
3/14/2025 – Exam 2
3/11/2025 – We answered questions ahead of the exam on friday and then started the next unit. We covered much of the content in the section on sets in the next set of handouts: Sets, Relations, and Functions Handout.
3/7/2025 – We finished off the material for unit 2. On Tuesday we will spen 20-30 minutes answering questions about the practice exam, if you ask them, then we will start the next unit. Exam 2 is on Friday.
3/4/2025 – We handed back the exams, handed out the redo exam, and the next practice exam. We discussed briefly finding closed formulas for sums and sequences. We will continue this on Friday.
2/28/2025 – We continued our discussion of sequences, covered summation and product notation, and started looking at particular types of summations.
2/25/2025 – Exam 1
2/21/25 – We spent a short amount to time going over the practice exam; remember in the future that if you want questions answered you need to ask them. We then started going through the packet on sequences and summations (Sequences and Sums Notes). We covered explicitly defined sequences and recursively defined sequences. Next class is the exam and then we will pick up on the packet next friday.
2/18/2025 – We more or less finished the logic packet and will start new material next class, after answering questions about the practice exam.
2/11/2025 – We looked at some more examples of recognizing valid argument forms and errors using truth tables. Then we looked at two examples of using valid argument forms to draw conclusions from premises. After that we looked at the first few examples of working with predicates and quantifiers.
2/7/2025 – We looked at material on conditional statements. We have finished chapter 2 in the text and will be starting chapter 3 on Tuesday.
2/4/2025 – Tomorrow we will start working through this packet of Notes on Logic. I’ll bring one copy for every one, but wanted to make sure you had access in electronic format as well.
1/31/2025 – We went over terminology from sections 1.1-1.3 in the text. We will start chapter 2 on Tuesday.
1/28/2025 – We continued reviewing various topics from algebra/precalculus as a warm up and we discussed some terminology related to sets and functions (these are in sections 1.2 and 1.3 in the text. We will cover everything we should need from chapter 1 on Friday.
1/24/2025 – We went over the syllabus and started our brief algebra review. We will review more algebra/precalculus on Tuesday and cover chapter 1 next Friday.

Unit 1: Logic

Writing Up Mathematics Assignment Due 2/7

Practice Exam for Unit 1 Due 2/21

Unit 2: Sequences and Summations

Practice Exam for Unit 2 Due 3/11

Unit 3: Sets, Functions, Relations

Practice Exam for Unit 3 Due 4/8

Extra Credit Exercise: The code below will build the equivalence classes from #26 in sec. 8.2. Explain how the code works both practically and based on the definition of the relation in 26.

# Written in Python
# Empty collection of equivalence classes
EC={}
# Build equivalence classes
for i in range(3):
    for j in range(3):
        for k in range(3):
            for l in range(3):
                # New element
                temp_lst=[i,j,k,l]
                # Assign to equivalence class based on sum
                EC[sum(temp_lst)]=EC.get(sum(temp_lst),[])+[temp_lst]
# Display equivalence classes
for key in EC: print(key,":\t",EC[key])

Unit 4: Counting, Combinatorics, and Graphs

Practice Exam for Unit 4 Due 5/9

Extra Credit:

Typesetting Out-of-Class Work (+5% on each typed up piece of work)

If you are doing this extra credit here is a quick reference sheet with symbols and such: Quick Reference

Create an account at Overleaf.com using your WCSU email account
Watch the introductory video here VIDEO LINK, the document created in the video is here: https://www.overleaf.com/read/khbktbvvvkjd
Follow this link to see samples of typed up exercises: https://www.overleaf.com/read/frptnykkgpbt
Follow this link to make a copy of a blank template: https://www.overleaf.com/read/hwdhcvmntrvs
Submit work by downloading a copy of the PDF you generate and printing it.

If you are having problems typing up a document and would like feedback you can share a link to it by following the directions here: https://www.overleaf.com/learn/how-to/Sharing_a_project; send me the link to view but not edit. You can see a full playlist of video lessons here: Technical Typesetting with \(LaTeX\) Playlist

Links and Handouts

Vocabulary

\(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), rational number, irrational number, set notation, set roster notation, modus ponens, modus tollens, De Morgan’s Law, negation, distribution, commutative law, associative law, \(\forall\), \(\exists\), converse, inverse, contrapositive, conjunction, disjunction, set, subset, Cartesian product, relation, function, sequences, series, summation, summation notation \[\sum_{i=0}^n a_i=a_0+a_1+\cdots+a_n,\] product notation \[\prod_{i=0}^n a_i=a_0\times a_1\times \cdots\times a_n,\] geometric sum \[\sum_{i=0}^n a\, r^i=a+ar+ar^2+\cdots+ar^n=a\, \frac{(r^{n+1}-1)}{r-1},\] union, intersection, set difference, set complement, power set – \(\mathscr{P}(x)\), reflexive, symmetric, transitive, anti-symmetric, function, one-to-one, onto, inverses, composition, addition principle, multiplication principle, possibilities tree, combinations, factorial, permutations, binomial coefficients, multinomial coefficients, graph, walk, circuit, Euler circuit, path, tree, trail, binary tree, Hamilton circuit, closed walk, … etc.

Dr. Charles Rocca

Professor of Mathematics