Foundational Discrete Mathematics

Syllabus

Class Calendar

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

  • …
  • 10/2 – Today we continued discussing sequences and we looked at geometric sums \[\sum_{i=0}^n a r^i=a\left(\frac{1-r^{n+1}}{1-r}\right)\] and arithmetic sums \[\sum_{i=0}^n mi+b=m\left(\frac{n(n+1)}{2}\right)+(n+1)b.\] We concluded by introducing recursively defined sequences like \(a_0=5, a_n=10 a_{n-1}+3\), we will look at these further next class. Looking at the practice exam for unit 2, you should be able to tackle problems 1-6, 9, and possibly 11; we will cover the rest of the material next class.
  • 9/28 – We handed back Exam 1, redos are due next Thursday. We then started discussing Sequences and Sums.
  • 9/25 – Exam 1
  • 9/21 – Covered a handful of topics from Chapter 3
  • 9/18: Today we reviewed material on logical arguments from chapter 2 and then discussed working with quantifiers, e.g. \(\forall\) and \(\exists\); this is the material from chapter 3. We will finish up material from chapter 3 on Thursday, but you already know enough to tackle the practice exam for unit 1. You should start the practice exam now if you have not already so that you have time to do it right and prepare yourself well for the exam.
  • 9/11 and 9/14: On these days we covered material from sections 2.2, 2.3, 3.1, and 3.2. In particular we discussed implications, converses, inverses, negations, and contrapositives, then we looked at valid and invalid argument forms, and we started discussing predicates, which are statements with variables e.g. \(E(x)\equiv x\ is\ even\), and their truth sets \(E=\{2,4,6,\ldots\}\). Lastly we finished class by looking at some universal and existential statements with predicates and discussing how we negate them: \[\sim\left(\forall x:E(x)\right)\equiv \exists x:\sim E(x),\] and \[\sim(\exists x: x>2\wedge x^2\leq 4)\equiv \forall x: x>2 \rightarrow x^2>4.\]
  • 9/7 – Today we looked at material from sections 2.1 and 2.2, as it says on the syllabus. We began by reviewing set roster notation, set builder notation, and quantifiers from last time. Then we discussed statements, negations (\(\sim P\)), conjunctions (ands, \(P\wedge Q\)), disjunctions (ors, \(P\vee Q\)), exclusive-or (xor, \(P\oplus Q\)), and implications (if-then, \(P\rightarrow Q\)). We looked at when each of these are true and how they compare (i.e. \(\sim(P\rightarrow Q)\equiv P\wedge \sim Q\)). We also discussed DeMorgan’s Law, Converses, Inverses, and Contrapositives.
  • 8/31 – Today we went over the syllabus and spent time discussing expectations for assignment submissions. Then we took a little time to introduce some basic terminology, in particular we discussed the natural number \(\mathbb{N}\), the integers \(\mathbb{Z}\), the rational numbers \(\mathbb{Q}\), the real numbers \(\mathbb{R}\), the complex numbers \(\mathbb{C}\), subsets, universal quantifiers \(\forall\), and existential quantifiers \(\exists\).

Unit 1: Logic

Practice Exam for Unit 1 Due 9-25-2023

WCSU Algebra Assessment in Blackboard Due 9/30 by 11:59pm

You must complete the Question Entry Practice and Algebra Pre-Assessment as part of your first assignment grade. You don’t have to write these up and your specific grade doesn’t matter. Please just complete them and do your best.

Assignment Due 10-2-2023

All text assignments must be written up neatly or be typed, and must be in complete sentences. Poor quality work can result in up to a 10% penalty.

  • Sec. 2.3 – 38bc
  • Sec. 3.3 – #’s 57 & 58;
  • Sec. 3.4 – # 34

Unit 2: Sequences and Summations

Practice Exam for Unit 2 Due 10-9-2023

Assignment Due 10-16-2023

All text assignments must be written up neatly or be typed, and must be in complete sentences. Poor quality work can result in up to a 10% penalty.

  • Sec. 5.1 – 87(Compare to Algorithm 5.1.1);
  • Sec. 5.6 – 18acd;
  • Sec. 5.7 – 54

Unit 3: Sets, Functions, Relations

Practice Exam for Unit 3 Due 10-30-2023

Assignment Due 11-6-2023

All text assignments must be written up neatly or be typed, and must be in complete sentences. Poor quality work can result in up to a 10% penalty.

  • Sec. 6.1 – 26;
  • Sec. 7.2 – 56;
  • Sec. 7.3 – 20;
  • Sec. 8.2 – 26

Unit 4: Counting and Combinatorics

Practice Exam for Unit 4 Due 11-20-2023

Assignment Due 11-27-2023

All text assignments must be written up neatly or be typed, and must be in complete sentences. Poor quality work can result in up to a 10% penalty.

  • Sec. 9.2 – 28;
  • Sec. 9.4 – 8 (look at Example 9.4.3);
  • Sec. 9.5 – 17 (try looking at a smaller problem first);
  • Sec. 9.7 – 16;
  • Extra Credit Sec. 9.3 – 26abcd;

Unit: 5 Graph Theory

Practice Exam for Unit 5 Due 12-7-2023

WCSU Algebra Assessment in Blackboard Due 12/15 by 11:59pm

You must complete the Algebra Post Assessment as part of your fifth assignment grade. You don’t have to write it up and your specific grade doesn’t matter. Please just complete it and do your best.

Assignment Due 12-15-2023 by 3pm (before the end of finals week)

All text assignments must be written up neatly or be typed, and must be in complete sentences. Poor quality work can result in up to a 10% penalty.

  • Sec 10.1 – 28 & 42;
  • Sec 10.4 – 24;

Final Exam Materials

Final Practice Exam Due 12-11-2023

Monday December 11th, Exam Period 3: 2pm – 4:30pm

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

  1. Create an account at Overleaf.com using your WCSU email account
  2. Watch the introductory video here VIDEO LINK, the document created in the video is here: https://www.overleaf.com/read/khbktbvvvkjd
  3. Follow this link to see samples of typed up exercises: https://www.overleaf.com/read/frptnykkgpbt
  4. Follow this link to make a copy of a blank template: https://www.overleaf.com/read/hwdhcvmntrvs
  5. 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

Writing Math Extra Credit Assignment (+2% to final grade)

Complete this packet on Writing Up Mathematics. This assignment needs to be typed.

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.