Syllabus


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, aternating. 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.

Unit 1: Logic

Practice Exam for Unit 1 Due Monday 9-20-2021

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

Practice Exam for Unit 2 Due Thursday 10-7-2021

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

Practice Exam for Unit 3 Due Monday 11-1-2021

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

Practice Exam for Unit 4 Due 12-2-2021

Assignment Due 12-9-2021 (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

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 \[\sum_{i=0}^n a_i,\] 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.