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.

  • …
  • 4-16-2024: Counting and Combinatorics Notes 1.0
  • 4-12-204: We covered the material on combinations with replacement in the Counting and Combinatorics Notes 1.0. Note, when looking at the for loops our calculation should have been \[{9+3 \choose 3} = {12 \choose 3} = 220\] because the 0 always had to be the first character in the string of characters. On Tuesday we will discuss the pigeonhole principle.
  • 4-9-2024: We covered matherial on factorials, permutations, combinations, and the binomial theorem in Counting and Combinatorics Notes 1.0
  • 4-5-2024: Exams were handed back, please try to get redos in in a timely fashion. We started discussing combinatorics with the multiplication and addition principles as well as decision trees.
  • 4-2-2024: Exam on Sets, Relations, and Functions
  • 3-26-2024: We discussed functions and completed enough that you should be able to complete the function questions on the practice exam and exam so we will go ahead with the exam on April 2nd. Remember that Friday is the day of reflection so there are no classes.
  • 3-19-2024 and 3-22-2024: We discussed relations and worked through the material in Sets, Relations, and Functions Handout (Version 1.0 Here). We will discuss the material on functions in the packet on Tuesday the 26th. Depending on how much we cover we may need to move the exam date.
  • 3-8-2024: Class Cancelled
  • 3-5-2024: We started the next unit today discussing sets and set notation. We made it through the top of page 4 in the Sets, Relations, and Functions Handout (Version 1.0 Here). You should try and read through the rest of page 4 and pages 5 and 6 before we meet again. Please recall that class is cancelled on Friday 3/8.
  • 3-1-2024: Exam 2
  • 2-27-2024: We went over an additional example of using the iteration method to find closed formulas for a first order recurrence relation and looked at some theorems on finding formulas for second order recurrence relations.
  • 2-23-2024: Today we worked through additional examples with arithmetic and geometric sums. We also introduced the iteration method for finding closed formulas for first order linear recurrence relations. We will pick up the discussion there next Tuesday. At this point you should be able to take a crack at problems 1-10 on the practice exam.
  • 2-20-2024: Today we reviewed a little bit about sequences and then looked at some examples of Arithmetic sums and Geometric sums. We’ll pick up there on Friday with the last couple examples in the summs section of the Sequence and Sums Notes (updated update, ver.1.3). At this point you should be able to make reasonable progress on problems 1-6 or 7 on the Unit 2 practice exam.
  • 2-13-2024: Class was again cancelled for a SNOW DAY. Here is a revised calendar for the semester: Revised Calendar
  • 2-9-2024: Exams were handed back, redo problems are due on 2-20-2024, they must be neatly written and in complete sentences. We covered sequences, we looked at the first four pages of this handout: Sequence and Sums Notes (updated update, ver.1.3).
  • 2-6-2024: Unit 1 Exam
  • 2-2-24: Today we went through the Tarski’s World Worksheet (revised version 1.1) in order to learn about truth sets, counter examples, and more about quantified statements. Remember, as it says on the syllabus, exam 1 is on Monday and you must turn in the practice exam when you come in for the test. As out of class work practice exams must be written up neatly and professionally just like the assignments.
  • 1-30-24: Today we did some examples of justifying valid arguments using other valid arguments; we used this Logic Laws Handout for reference. You will be given an abbreviated version of the handout for the exam; you need to know the fundamental terms for negation, conjunction, disjunction, implication, contrapositive, converse, inverse, negation, modus ponens, modus tollens, converse error, and inverse error. After this we started looking at predicates (statements with variables) and quantified statements. At this point you should be able to do problems 1-4, 6, 7, and 8 on the practice exam, and can probably take a crack at the others. We will finish up the remaining material on Friday.
  • 1-26-2024: Today we discussed different ways of writing implications, valid arguments, and invalid arguments. This means we have finished the basics from chapter 2 which we need to cover.
  • 1-23-2024: Today we went over the syllabus, discussed proper formatting for out of class work, and then we talked about statements, negations (~, not), conjunctions (\(\wedge\), AND), disjunctions (\(\vee\), OR), exclusive ors (\(\oplus\), XOR), and implications (\(P\rightarrow Q\), If P, Then Q). We looked at the truth values of all of these using truth tables. This also gave us our first logical rule, DeMorgan’s Laws: \[\sim (P\wedge Q)\equiv \sim P \vee \sim Q\ and\ \sim (P\vee Q)\equiv \sim P \wedge \sim Q.\]

Unit 1: Logic

Practice Exam for Unit 1 Due 2-6-2024

Assignment Due 2-13-2024 2-20-2024

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 – # 32

Unit 2: Sequences and Summations

Practice Exam for Unit 2 Due 3-1-2024

Assignment Due 3-8-2024

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 4-2-2024

Assignment Due 4-9-2024

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 4-19-2024

Assignment Due 4-26-2024

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 5-3-2024

Assignment Due 5-9-2024 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 5/7/2024

The final exam for this class is at 2pm on Tuesday May 7th.

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.