MATH 240 Introduction to Mathematical Thought
Spring '99      Class Summaries
 1/20 1/22 1/25 1/27 1/29 2/1 2/3
• 1/20 First class was spent going over the structure of the course. We did look at one example of a proof without words to see how   x+1/x >= 2    for any positive x.
• 1/22 The discussion focused on the two concepts from Solow: Unification and Generalization. We considered many examples of unification related solving equations such as 3x=12, 3x=24; then 3x=7 and 3x=N. We considered methods to solve these... guess and check,  list and search,  symbolic recognition of patterns, and algebraic solution using analysis. This led to being able to also work with an equation of the form 5x=4 and to a discussion of the solution to the unification in Ax=B being solved by x= B/A.

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We looked at another context to examine Unification and generalization. First we looked at showing the if f(x) = x2 then f '(3) = 6 using limits and the definition of the derivative. We moved directly to a generalization that f '(a) = 2a which could be demonstrated directly again using the definition of the derivative and limits. we discussed how one could do similar work to show that if f(x)= x3 then f '(x) = 3x2 and one could arrive at a unification in the statement that if n is a natural number  and f(x) = x then f '(x) = nxn-1. If we think of n as a negative integer, say -4, then this is an example of generalization because the statement of the result extends to instances which are not covered by the common experience with 2 and 3. Likewise we can genralize the result to f(x) = xr where r is any  fraction or more generally any real number, but a proof of that result might involve much more complicated arguments.

Next class: Some fundamental discussion of sets.... perhaps abstractions.

• 1/25 The  discussion started with an examination of some of the problems assigned from Solow. These used the concept of generalization  and unification to consider how expressions could be generalized.

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The discussion then turned to a beginning look at this week's assignment of a Proof without Words 1 demonstration of the formula: 1 + 2 + 3 +...+ n =  n2 /2 + n/2. After partners had a chance to discuss the figure and its relatio to the formula, the proof organization was discussed in some detail, emphasizing the need to describe the figure in relatio  to the general formula.
Next class each student is asked to bring a draft of the proof that the partners can review and discuss prior to friday, when the proof will need to be submitted.

The discussion then moved to the nature of sets, noting especially that for a set to be well defined it must be possible to determine when some object is or is not a member of the set. Some important sets that we will consider this term are as follows:

 the natural numbers N {1,2,3,...} or {0,1,2,3,...} the integers Z {0,1,-1,2,-2, ... } the rational numbers Q { n/m where n and m are integers and m not 0} the real numbers R { numbers that can be represented as decimals}
The issue of being a member of  a set can be an issue of the nature of the set or the nature of the object. For example, we can consider sqrt(2) and ask, "is this a real number?" "Is this a rational number?" To determined if sqrt(2) is a member of R,we must have a way to determine this number as a decimal, that is to determine what digit would be in th ekth decimal place for any k. Now this is possible through some estimaton procedures, though there is no closed form solution to determine the kth decimal place of the sqrt(2). On the other hand the sqrt(2) is not a member of the rational numbers. This is not because the rational numbers have a problem with the definition of that set, but is a question of mathematics to demonstrate that this number cannot be expressed as a ratio of integers as required by definition for membership in the set of rational numbers.

Next time(?): More on sets, membership, subsets, unions, intersections, complements, generalization, abstraction, and proofs of conditional statements.

• 1/27 Note: The assignments in SOS were made assuming the work was being done in the first (1964) edition. The assignments will be revised to conform to the 2nd(1998) edition in the next few days.

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The discussion today focused on the nature of the subset relation. The initial discussion  focused on  the importance of recognizing the universal set involved in a discussion. Thus for example the set {x: 0 = 10 + 2x + x2} is an empty set if the universe is the real numbers, but contains two members if the universe is the set of complex numbers, C.

We turned our attention to some examples of sets of functions... C0 = { continuous real valued functions defined on all the real numbers} and C1 = { differentiable real valued functions defined on all the real numbers}. The issue was whether these sets are equal. This led to as discussion of the subset relation : when set A is a subset of a set B. The definition of this term can be made in the form of an absolute statement: "every element of A is an element of B" or as a conditional statement: " if f is an element of  A then f is an element of B". We observed that C is a subset of C0. To prove this we suppose that f is a member of C1and thus f is a differentiable function. It is a result of the calculus course that if a function is differentiable, then it is continuous. Thus f is a continuous function, which means it is a member of  C0 .

To consider the related issue, we asked "Is C0  a subset of C1 ?"   Here the answer is : NO. To show that this is true we examined the function f (x)= |x|. This function is continuous for all real numbers, but is not differentiable at x = 0. This gives an instance where the conditional requirement for a subset fails, since the hypothesis is true for this function but the conclusion is false.  The example also shows why the absolute statement is false by giving a single member of C0 which is not a member of  C1 . This lead to a discussion of the truth or falsity of conditional  and universal absolute statements in mathematics.

For next class we will continue a discussion of sets while also considering some aspects of the logic we use to deal with mathematics and proofs. The Proof without Words I is also due on Friday.

• 1/29  Today I finally got a hold of the 2nd edition of the Shaum's Outline.... It has made some major changes from the last edition.... The assignments will be revised in the next few days.  :(.

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Today's class reviewed how to prove sets are equal using A={ 2, -2} and B = { : x= 2}.
(i) First, A is a subset of B. This is demonstrated by checking the equation of B for x = 2 and x = -2.
(ii) Then it is neceassary to show that B is a subset of A. For this we suppose x is an element of B, so that x= 2, and then x2 - 2 = 0, so (x +2)(x-2) = 0. considering the last equation, together with the fact  that if the product of 2 real numbers is 0, then one of the factors is 0, we see that either x = -2 or x = 2. In either case, is a member of A.

The class continued with an examination of various set operations... intersection, union, difference, and complement. The definitions of these operations were illustrated with some finite sets and Venn diagrams.  The class concluded with a statement of a set equality: (AuB)c = Ac intersect Bc. We will discuss the proof of this next week.

• 2/1 Today we started with a discussion of a homework problem related to generalization.  The first problem generalize from a quadratic expression in x to a general real valued function of x. The second problem (more difficult) generalized a specific linear inequality in two variables to a more general linear inequality in two variables, and then to three variables, and finally to n - variables. Also under this problem was the relation of (0,0) to the inequality and how that would be involved in the generalization.

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We then discussed how simple statements appeared in compound statements using "and", "or", "if...then", and "not" and how to determine the truth of the compund sentence from knowing the truth of the simple statements. We examined wht there were 4 possible ways to have truth values for a pair of simple statements and how these would deteremine the truth value of a compund statement using a "truth table."

 A B A and B If A then B T T T T T F F F F T F T F F F T
This discussion will be continued next class.

• 2/3 The discussion focussed again on finding the truth values for complex statement using conjunction (and), disjunction (or), conditional (if..then), and negations (not).

• A proposition is called a tautology if it is true without regard to the truth or falsity of any of its component statement.

A proposition A is said to be equivalent to another proposition B is the truth value of A is the same as the truth value of B for any choice of truth values for its primitive components.

We also considered the connective "if and only if", <->, called the biconditional. It has the following truth table:

 A B A <->B T T T T F F F T F F F T
Proposition: The proposition  A is equivalent to the proposition B when and only when the proposition A<->B isa tautology.

Next time: Arguments.

• Discontinued..... Sorry..... Don't miss class ... it won't be here anymore....