MATH 240 Introduction to Mathematical Thought
Fall '98      Class Summaries
 8/26 28 31 9/2 4 7 no class 9 11 14

• 8/26  We started the class by people exchanging their work on the assignment Proof w/o Words #1.

• Each student read another student's explanation of the first proof without words, asked questions to help the understand the explanation and then read the proof back to the author. We then discussed how individuals and started and ended their explanations, and some of the ways people had used algebra and geometry in the body of their explanations.

We began a discussion of sets by considering two examples:  S={1,3,5,7} and T=(red,blue, yellow} and the issue of what it means for a set to be "well defined." In particular for a set to be well defined one must be able to determine whether something is a member or not of the set. Thus 3 is a member of S, 2.999 is not a member of S, and 2.9999... is a member. ( 2.999... is another way to denote the number 3, so we distinguish the number 3 from the way it is represented with numberal or using decimals.)
[ Let A = 2.99... . Then 10A= 29.9..., and hence 9A=10A-A=27. So A = 27/9 = 3. ]

Next time:  More on sets.

• 8/28  Next class we will change rooms to Theater Arts 011.

• The main part of the class was spent discussing Problem 1.7 in D&W. The emphasis was on trting to follow some of Polya's suggestions in How to Solve It in organizing an approach to eventually obtaining a "proof". By "rpoof here we mean primarily a convincing argument that may explain the claimed result.  After some discussion about what the problem sought, we drew some figures to help us visualize the problem. Further work with these visualizations allowed us to refine what we thought would help solve the problem. Eventually we had an outline of the key steps needed to develop a proof and the remainder of the task was left for the students to complete in writing the arguments needed to transform the work done in class into a proof.

Here are some of the figures we used to think about the problem. (still to be done....)

We spent a little time at the end of class on the nature of  work that uses sets... in particular the importance of recognizing the context (or universe) in which a disucussion using sets transpires. The contexts of mathematics related to numbers were connected to various levels of school work.

 counting numbers primary grades fractions elementary grades negative numbers pre-algebra rational numbers pre-algebra real numbers sqrt(2) and pi geometry and algebra imaginary and complex numbers algebra

We also distinguished numbers from the notations used to represent them,  e.g., which is larger 3 or 5 ? and the symbol for pi doesn't tell you anything about the nature of this number, while sqrt(2) does indicate that this number when squared equals 2.

Next class: More on sets! and TA 011.

• 8/31 After announcing the assignment for Wednesday, Problem 1.15a in D&W was designated as Problem of the Week. The two proofs without words that are due on Wednesday were also discussed in some detail, with the organization of the figures and the key elements that might warrant more explanation being highlited.

• We continued the discussion of sets and the notation. Lists, verbal descriptives, and list patterns were illustrated with examples showing again the need to be aware of the context of a universal set and to be explicit about patterns that are sometimes presumed. Set equality was noted as depending on the ability to identify set membership and to be able to justify that the elements of each set are members of the other, i.e.,
Def'n: If A and B are sets, we say  the sets A and B are equal, A=B, when every element of A is an element of B and every element of B is an element of A.

We then turned our attention to some elementary set operations: Union, intersections, subtraction, and complement.Each of these was illustrated with an example where the sets were given by lists.

Next time: More on sets... and a beginning to some more details on the nature of proof.

• 9/2. After announcing the assignment for Friday, the next Proof w/o Words #3 was distributed and discussed. This is a proof of an equation which uses the definitions of the arctangent and some basic facts about right triangles.

• Back to a discussion of sets, we introduced the relation between sets of one set being a subset of another, and saw how we could use the definition to examine when this relation is not true and when it is. [I have more to write here!] An example of some sets given by lists, and one where the sets are given by decriptions using inequalities.

• 9/4&9/9. Sorry :( .  I didn't get to writing summaries for these classes.  We continued to work on proofs involing sets. Set inclusion and set equality. We started work on the cartesian product of sets.
• 9/11. We spent much of the class time on a proof of set equality. We looked at the diagram for the equality first to see that the equality appeared correct. We then proceeded to look at a proof using the definitions of the various set operations. In the proof we recognized how sometimes it is convenient to work backwards from a goal in contrast from working forward from an assumption. When working backwards it is important to be clear that the goal of the proof is being transformed and to be able to  justify why achieveing the new goal is adequate for completing the proof. We also briefly discussed another apporach to porving set equalities using an algebra of set equalities including such equalities as DeMorgan's identities. This approach was discouraged at this stage because although the proofs are correct, effficient and almost mechanical, this style of proof does not involve examining the nature of the actual sets and the elements of these sets. In many mathematical setting, it is important to be able to move from the set to the element level in an argument.

• At the end of class we started further discusion of Cartesian products of sets. In particular we discussed how to visualize products using horizontal and vertical features to locate points in a picture of a product.
Here is an example:

 A AxB 8 (1,8) . . . (8,8) 5 . . . . . 4 . . . . . 3 . . . . . 1 (1,1) (3,1) (4,1) (5,1) (8,1) 1 3 4 5 8 B
• 9/14 Much of the class was spent discussing the use of  Ax Ey format in statements. Besides the example in the assigned problems which characterize continuous and uniformly continuous functions, we considered the statements:

•       (1)  For any a there is a number b so that a+b=0.
(A a)   (E b)  (a+b=0)
(2)  There is a number b so that for any a a+b=0.
(E b)   (A  a)  (a+b=0)

Statement 1 is true, while statement 2 is false.

We compared (1) and (2) with the similar statements:
(1)  For any a there is a number b so that a+b=a.
(A a)   (E b)  (a+b=a)
(2)  There is a number b so that for any a a+b=a.
(E b)   (A  a)  (a+b=a)
In this case both statements are true, because we can use the number 0 for b.

Thinking about the examples we saw that in any case where  there is a switching of universal and existential quantifiers in their relative position in the logical structure of a statement, the statement of form 2 will imple the statement of form 1.
Informally this means that if you have some b that works for all a, you certainly can use that b to respond to the need to show the existence of a b that would work for any specific a.

We next looked at determining the truth of compound statements and how this can be determined by the use of "truth tables" We constructed  the basic truth tables for ~A, A -> B, A & B,  A v B, and  A<->B, and examined briefly an example of using these to determine a truth table for a mor complicated compound statement. Here's what a truth table for the basic connectives looks like:

 A B ~A A->B A&B AvB A<->B T T F T T T T T F F F F T F F T T T F T F F F T T F F T