Statements, theorems, and axioms

To read: Straight and Wilson pp. 1-6; stop before the last paragraph on p. 6

Notes : The most important things are the definitions of conjecture, theorem, argument, propositional function (which we will call "property"), and statement. However, the material on perfect numbers and primes serves as an excellent motivation for these concepts and allows for some nice illustrations of them.

Reading questions:
1. Give an example of an odd prime number which is not a Mersenne prime.
2. Is the following a statement? "It will snow in Dunkirk on Friday."
3. Write the logical negation of the statement "Every student completed the reading assignment." (No explanation is necessary.)
4. Is 36 deficient, perfect, or abundant?
Bonus. What is the difference between a statement and a propositional function?

Set notations

To read: Straight and Wilson pp. 6-9; start with the second paragraph on p. 6. (This has some overlap with the 1-25 reading assignment.)

Notes : We will discuss the theory of sets in depth later. The important thing here is to learn how to describe sets. Certain common sets are denoted by special symbols. You should know these. Additionally, a set might be denoted by A) listing its elements, B) giving a rule that determines the elements, or C) in the case of sets of real numbers, using interval notation. Know and use proper format for each type of notation.

Reading questions:
1. How many elements are in the set {5, 6, 7, . . . , 21}?
2. Is 0 an element of Z⁺?
3. Let C be the set of all cities in the United States, and let S be the set of all states in the United States. Is the statement "C is a subset of S" true or false?
4. Let S=(-3,7]. Write the set S^- in interval notation.

Quantified statements

To read: Section 7 (pp. 3-4) of Proof techniques in a nutshell (Cox)

Notes : This reading briefly describes the two main types of quantified statements and provides a summary of methods used to prove and disprove them. We will spend the next few days going over this material in more detail, but I hope this section will be a useful introduction and a handy reference. Pay special attention to which goals involve working with arbitrary elements, and which just require one example. It is very important to be able to distinguish these!

Reading questions:
1. Is "The polynomial x² + 2x + 1 has a real root." universally or existentially quantified?
2. Give a counterexample to the universally quantified statement "In English, every month has a letter 'r' in its name."
3. According to the guidelines in the reading, what would be an appropriate first sentence for a proof the following statement?

Quantifiers

To read: pp. 24-28 (in the text); through the Solution of Example 1.15

Notes: Quantifiers provide a very common and useful way to turn propositional functions into statements. Note there are two quantifiers; you should know their names and symbols. The emphasis here is on gaining familiarity in working with the quantifiers, particularly in translating quantified statements from natural language to symbolic form and vice versa.

Reading questions:
1. If a propositional function is modified by inserting a quantifier in front of it, must it necessarily become a statement? (Hint: Consider the analagous question with plugging in a specific value for a variable instead of inserting a quantifier.)
2. In the process of following the reading's suggestion for showing part (b) of Example 1.14 is true, what integer would you need to show is prime?
3. Is the assertion in part (b) of Example 1.15 true or false? Explain how you know.
Bonus. Is there a smaller positive integer than n=36 that could be used to show Example 1.14(b) is true? Explain.

To read: NOTHING

Notes: There is no reading assignment for this day.

Negations of quantified statements

To read: Straight and Wilson pp. 28-29; start after Example 1.15

Notes: Being able to negate and simplify quantified statements is one of the most important basic skills for this course. This tactic will often help us see how to proceed with a proof. The reading uses some logic symbols we haven't talked about yet. You might be able to pick up on what these are, but focus on what happens to quantifiers under negation.

Reading questions: In 1 and 2, provide the simplified negations of the given statements. (Here simplified means that negations should only apply directly to simple predicates, not to anything compound or quantified.) Hint: Translate to symbolic form, negate the formula, and then translate back to get the natural language negation. (But you do not have to show this work.)
1. There exists a positive integer n such that for every prime number p, p is less than or equal to n.
2. For every positive integer M, there is a real number x such that 1/x > M.

Arguments, validity, form

To read : pp. 7-9 in How to Prove It by Velleman (on ANGEL); Stop before Example 1.1.2

Notes : This is where we begin learning more generally how to construct proofs for a wide variety of more complex types of statements. Logic is the fundamental tool. The key concept here is validity of an argument. Notice how it depends only on the form of the argument, not on its specific substance. We will reveal the form of an argument by concentrating on a few crucial words, which we will convert into special symbols.

Reading questions:
1. Is the following argument valid? (Explain.) You may assume r is properly bound.

r is greater than or equal to zero.
It's impossible for r to be greater than zero.
Therefore, r equals zero.

Analyzing logical form

To read: pp. 9-11 in How to Prove It by Velleman (on ANGEL); Start at Example 1.1.2

Notes : This reading illustrates how to put into practice the strategy of converting natural language statements into symbolic form (and vice versa). Doing so quickly brings up some technical issues that occur frequently, and Velleman addresses them. Pay particular attention to the use of parentheses, the rules for well-formed formulas, and the imperfection of the correspondence between English words and connective symbols.

Reading questions:
1. Let e represent "MATH 210 is easy" and f represent "MATH 210 is fun." Convert the following formulas to English, expressing their meanings as clearly and naturally as possible.
    (a)  ~ e ∨ (f ∧ e)
    (b)  ~ (e ∧ f) ∨ e
2. Convert the following statement entirely to symbolic form, assigning a letter to each simple sentence involved: "The shelter is cold, but dry."
3. "Mitt Romney or Rick Santorum will win the Republican nomination." In this statement, is "or" used as a logical connective (i.e., to combine two simpler statements)?

Connectives (not, and, or)

To read: Straight and Wilson pp. 10-11; go through the Solution of Example 1.2

Notes : Connectives are essentially defined by their truth tables. You should know the truth tables for not, and, or. You should also know the symbols for each of these connectives.

Reading questions:
1. If p is true and q is true, what is the truth value of the disjunction p ∨ q?
2. Give the negation of the statement "The sky is dark or Jon is awake." Write your negation in the simplest form possible, with negatives (if necessary) only on the simple components.
Bonus. If a formula contains 7 distinct variables, how many rows will its truth table have? Explain! (Tip: Do NOT try to actually construct such a table and count its rows.)

To read: NOTHING

Notes: There is no reading assignment for this day.