## Final Exam Week Office Hours:

• Tuesday 12-7-2021: 2:30-4:30pm
• Thursday 12-9-2021: 1:00-3:00pm
• Monday 12-13-2021: 12:00-2:00pm
• Tuesday 12-14-2021: 3:00-5:00pm

## Class Calendar

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

• 8-26-2021: Today we covered the syllabus, talked about the structure of the class, and talked about writing up math problems; the slides on writing up problems are posted below with the links and handouts.  We then started looking at chapter one by discussing some terminology related to set (see section 1.2).  P.S. For those who are interested MAT 186: Technical Typesetting with $$\LaTeX$$ (CRN 10819) is the class on typing up technical documents which I mentioned.
• 8-30-2021: Today we finished covering some introductory vocabulary from chapter 1; all the individual topics will be revisited this just gave us some common language. We then started discussing material from section 2.1 on statements. We touched on Negation (NOT, $$\sim P$$), Conjunctions (AND, $$P\wedge Q$$), Disjunctions (OR, $$P\vee Q$$), and Exclusive Or (XOR, $$P \oplus Q$$). We also defined tautology and contradiction.
• 9-2-2021: Today we covered material from sections 2.1 and 2.2. In particular we looked at logical equivalences, these are summarized on the table I handed out (a copy of which is below) and on table 2.1.1 p. 35 in your textbook. I also added a copy of the blank truth tables to the links and handout section below. We briefly discussed Valid and Invalid Inferences. Then we discussed conditional statements, $$P\rightarrow Q\equiv \sim (P\wedge\sim Q)$$, and their negation, $$P\wedge \sim Q$$. For example “If I am carrying an umbrella, then it is raining.” versus “I am carrying an umbrella and it is not raining.” We will pick up here next class.
• 9-9-2021: Today we covered material from sections 2.2 and 2.3 on conditionals and valid/invalid arguments. This included: conditionals, contrapositive, converse, inverse, negation, elimination, Modus Ponens, Modus Tollens, converse error, inverse error, contradiction, cases, and transitivity. The valid arguments are summarized on the green sheet I gave out last class and on Table 2.3.1 on page 61 of your text. We ended by looking at problem 43 from section 2.3 to see how these ideas could be used together. You should be able to start working through the practice problems from chapters 1 and 2. We will jump right into chapter 3 next time discussion quantified statements and playing with Tarski’s world and this handout. Take a look at 3.1 and start playing with the handout if you want to get a head start.
• 9-13-2021: Today we started by looking at two examples of using our rules of inference from chapter 2 to justify a conclusion given a set of premises. For the first example we were only given the premises and had to find the steps and justifications; emphasis was placed on examining the premises and conclusion first before we start applying rules so that we could make a plan of attack. For the second example we were given all the steps and focused on giving reasons; this was a proof by contradiction which has the form $$(P\wedge \sim Q\rightarrow Contradiction)\equiv (P\rightarrow Q)$$. Then we started chapter 3 where we discussed universal and existential quantifiers and there use with predicates. For the last part of the class we looked at the Tarski’s World Handout. After doing a couple examples together I asked you to look at the first three problems on the back and for each one write down two truths and a lie; we will look at these next time.
• 9-16-2021: Today we played with quantified statements and their negations using the Tarski’s World handout; more examples of this are in your text in sections 3.1 and 3.3. We also used these examples to understand statements with multiple quantifiers such as $$\forall x\, \exists y\, : \ldots$$ or $$\exists x\, \forall y\, \ldots$$. Finally, we looked at the material from section 3.4 on inference with predicates and quantified statements. We saw that for the most part this was just a generalization of what we saw in chapter 2 when we discussed contrapositive, converse, inverse, negation, Modus Ponens, Modus Tollens, Converse Error, Inverse Error, and Transitivity. We also played with the idea of visualizing logical inference in terms of sets, i.e. we can visualize the statement $$\forall x:\, P(x)\rightarrow Q(x)$$ as saying that the truth set for the predicate $$P(x)$$ is a subset of the truth set for $$Q(x)$$. This ended the material for unit one; the unit 1 exam is on Monday the 20th and you must turn in your unit 1 practice exam when you come in for the exam.
• 9-20-2021: Today was exam 1
• 9-23-2021: Today you got the test and practice test back. At the top of the test is a list of questions you need to redo for the redo portion of the exam which is worth 5% on its own and can earn you back up to 30% of the points you lost. The redos are due on 9/30/2021 and they are out of class work and so need to be written up as such. After getting back the exams we started the next unit. We covered sequences and some adjectives for them: increasing, decreasing, bounded, unbounded, monotonic, alternating. We also discussed summations and we will pick up with those on Monday. At this point you already know enough to try problems 1,2,3,5,7,8, & 9 on the Unit 2 Practice Exam. Also, recall that your first assignment is due monday.
• 9-27-2021: Today we continued talking about summations. We covered geometric, arithmetic, and telescoping sums and how to evaluate them: $\sum_{i=0}^{n}a\, r^i=a\left(\frac{r^{n+1}-1}{r-1}\right),\ \sum_{i=1}^n i =\frac{n(n+1)}{2},\ \text{and}\ \sum_{i=1}^{n}\frac{1}{i}-\frac{1}{i+1}=1-\frac{1}{n+1}$ Toward the end we introduced products and looked at properties of sums and products. This material was still from section 5.1. Next time we will be getting into 5.6 and 5.7.
• 9-30-2021: Today we looked at recursive sequences such as $$m_n=2 m_{n-1}+1$$ and how we could write them in a closed form by using iteration like $m_5=2\, m_4+1=4\, m_3+2+1=8\, m_2+4+2+1=16\, m_1+8+4+2+1=16+8+4+2+1=2^5-1.$ This process works well for recursive expressions like the one above but we will see that for expressions like the Fibonacci sequence, $$F_n=F_{n-1}+F_{n-2}$$, we will need more sophisticated techniques.
• 10-4-2021: Today we began by practicing finding closed forms for recursive sequences of the type $$a_n=A\, a_{n-1}=B$$ using iteration. In particular we found a closed formula for calculating the amount of money you could save given regular annual deposits and a fixed yearly interest rate. Then we looked at how to find a closed form for recursive sequences like the Fibonacci sequence, i.e. those of the form $$a_n=A\, a_{n-1}+B\, a_{n-2}$$ which are called Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients (sorry I forgot the constant coefficients part in class). In particular we found a closed form for the Fibonacci sequence using Theorem 5.8.3 on p. 321. This theorem assumed the characteristic polynomial, $$t^2-At-B$$, had two distinct roots; for only a single root we would use Theorem 5.8.5 (try looking at this for extra credit on the exam). Don’t forget that your exam is Thursday and the practice exam is out of class work and so must be neat, in sentences, and it should be clear what you are answering for each question.
• 10-6-2021: Exam 2
• 10-11-2021: Today we handed back the test, the exam redo’s are due next Monday, October 18th. After that we started new material from section 6.1. We covered a lot of definitions including: roster notation, builder notation, set product, universal set ($$\mathcal{U}$$), empty set ($$\emptyset$$), power set ($$\mathcal{P}(x)$$), union ($$A\cup B$$), intersection ($$A\cap B$$), compliment ($$A^c$$), and set minus or difference ($$A-B$$ or $$A\setminus B$$). You know enough to start looking at the chapter 6 material on the unit 3 practice exam. Remember that your second assignment is due Thursday. Also, as it says on the syllabus and as we have discussed in class, the quality of your work matters; you need to be neat, to write in sentences, and to make it clear what the original question was without just repeating it. We have done multiple examples of this including the presentation the first day of class (Writing up Math Problems) and there is an entire extra credit assignment demonstrating this (Writing Math Extra Credit); there is no excuse at this point for not at least trying to follow these guidelines.
• 10-14-21: Today you got back the practice exam; remember that the exam redos are due Monday 10/18/12 and need to follow the same guidelines as all out of class work. We then covered more material on sets, section 6.1, and then started on material from sections 8.1 and 8.2 on relations. At this point you should be able to make a fairly good effort on the chapter 6 problems on the practice exam; for question 3 approach it the way you did questions about logic from the first unit using the Sets Reference Table. You should also be able to read some of the practice material from chapter 8 and start thinking about it.
• 10-18-2021: Today we covered more examples of relations and introduced the idea of an inverse relation. We looked at how a relation could be represented by a Directed Graph. We then discussed equivalence relations which must satisfy all three of the properties of being reflexive, symmetric, and transitive. Finally we looked at how equivalence relations split up sets into disjoint subsets called equivalence classes which form a partition of the entire set. We have covered the vast majority of material from sections 8.1, 8.2, and 8.3 and you have all the basic knowledge needed to answer the chapter 6 and chapter 8 questions on the practice exam. Thursday we will look at modular equivalence which is a particularly important example of an equivalence relation and then cover some basic material from section 8.5 before moving onto chapter 7.
• 10-21-21: Today we discussed equivalence relations more and introduced antisymmetric relations, partial orderings, and total orderings from section 8.5; we saw that we could represent these with Hasse diagrams. Then we jumped to section 7.1 and introduced functions, domain, codomain, range and image.
• 10-25-21: Today we covered section 7.2 discussing functions which are one-to-one, onto, invertible, and bijections. We discussed these both in terms of arrow diagrams and graphs. We finished by looking at the Hamming Distance function and its relation to the XOR logical operator ($$\oplus$$).
• 10-28-21: Today we spent more time going over 1-1 and onto functions. We saw that if you compose two 1-1 functions the result is 1-1 and likewise if we compose two onto functions the result is onto, but if we compose a 1-1 function with an onto function we don’t know that it will be either. We also looked at an example where we had to restrict the domain and codomain of a function so that we could compose it with another function. Essentially what we saw is that if $$f:R\rightarrow S$$ and $$g:T\rightarrow U$$ then $$g\circ f$$ will only be well defined if $$S\subseteq T$$. We ended class discussing the cardinality of sets in particular we saw that $$\mathbb{N}$$, $$\mathbb{Z}$$, and $$\mathbb{Q}$$ are the same size infinity, but $$\mathbb{R}$$ is a bigger infinity.
• 11-1-21: Exam 3
• 11-4-21: Today you got back exam 3, the redos are due next Thursday, 11-11-21. As you do the redos be sure to look at the comment slides from today’s class: Click Here For Slides. After talking about the test we started looking at Section 9,1 on probability and the multiplication principle. There is a handout of notes on probability below in the Links and Handouts section. We will pick up next class with Sections 9.2 and 9.3.
• 11-8-21: Today we recapped some of 9.1 and discussed sections 9.2 and 9.3. At this point we have discussed probability, sample space, outcome, event, disjoint and independent events, the multiplication principle, possibilities trees, and the inclusion/exclusion principle. For the inclusion/exclusion principle we saw that $|A\cup B|=|A|+|B|-|A\cap B|$ and $|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|C\cap B|+|A\cap B\cap C|.$ We also looked at the connection between the inclusion/exclusion principle and power sets; given a set of sets $$X$$ we could write $\left|\bigcup_{s\in X} s\right|=\sum_{S\in\mathscr{P}(X)}(-1)^{|S|+1}\left|\bigcap_{s\in S} s\right|.$ You should be able to make a fair attempt at problems from sections 9.1, 9.2 and 9.3, and try problems 1-4 on the unit 4 practice exam.
• 11-11-21: Today we started class by going over examples similar to the assignment that was handed back. I particularly emphasized the importance of trying examples, using visualizations, and making sure you understand the problem before trying to give an answer. Then we covered the pigeon hole principle and its generalization from section 9.4. We will do a couple more examples at the start of the next class and then cover section 9.5.
• 11-15-21: For the first half of class we looked at some additional examples related to the pigeon hole principle. We then started on section 9.5 and combinations and permutations. We will do additional examples from 9.5 on Thursday as well was starting on 9.6. At this point you should be able to try problems 1-4 and 9-11 on the Unit 4 practice exam and possibly have a go at 5 and 6.
• 11-18-2021: Today we discussed combinations with repetition, i.e choosing $$k$$ objects from $$n$$ when you can pick a particular object more than once. In the text, this material is in section 9.6. At this point you should be able to make a reasonable attempt at problems 1-11, 13, & 15 on the Unit 4 Practice Exam. We will do a couple more examples from section 9.6 on Monday and then finish off the unit with material from section 9.7.
• 11-23-2021: Today we looked at more examples of combinations with repetition including one example that required the use of the inclusion exclusion principle. We then started looking at examples from section 9.7 involving the Binomial Theorem. We will do more of these next Monday, but at this point you already know enough to complete the practice exam and study for the last unit exam on December 2nd. You should also be able to start picking away at the practice exam for your final.
• 11-29-2021: We finished up material on the Binomial Theorem and looked at Pascal’s Formula ${n \choose k} ={n-1 \choose k}+{n-1 \choose k-1}.$ Thursday is your fourth exam and next Monday is a review class so please come prepared with questions. The final exam is on Monday December 13th between 2 and 4:30.
• 12-2-2021: Exam 4
• 12-6-2021: Review
• 12-13-2021: Fanal Exam at 2pm

## Extra Credit:

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

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

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

## Unit 1: Logic

### Assignment Due 9-27-2021(all page numbers refer to the 4th edition)

• p.38 – #52;
• p.49 – #’s 20beg, 22beg, 23beg;
• p.116 – #’s 7, 8, 15, 37;
• p.144 – #’s 33 & 34

## Unit 2: Sequences and Summations

### Assignment Due 10-14-2021 (all page numbers refer to the 4th edition)

• p.242 – #’s 15, 52, 39;
• p.302 – #’s 14, 18abc;
• p.314 – # 8 (use the formula from #2)

## Unit 3: Sets, Functions, Relations

### Assignment Due 11-8-2021 (all page numbers refer to the 4th edition)

• p.351 – # 26;
• p. 414 – #’s 9, 11;
• p.427 – #’s 22, 24;
• p.458 – #’s 22, 29

## Unit 4: Counting and Combinatorics

### Assignment Due 12-9-2021 12-13-2021 at the final (all page numbers refer to the 4th edition)

• p.538 – #’s 22, 29;
• p.549 – # 23;
• p.564 – #’s 8 (look at 7 for a hint), 32 (builds on 7 & 8);
• p.582 – # 17 (try looking at a smaller problem first);
• p.591- # 18 (compare to 10, 11, & 16);
• p.604 – # 41

## Final Exam Materials

### Final Practice Exam Due 12-13-2021

Monday December 13th, Exam Period 3: 2pm – 4:30pm

$$\mathbb{N}$$, $$\mathbb{Z}$$, $$\mathbb{Q}$$, $$\mathbb{R}$$, 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 $\sum_{i=0}^n a_i,$ product notation $\prod_{i=0}^n a_i,$ geometric sum $\sum_{i=0}^n a\, r^i=a\, \frac{(r^{n+1}-1)}{r-1},$ 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, … etc.