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) = xⁿ  then f '(x) = nx^n-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) = x^r 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.
 

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:

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

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 C¹  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.

Today's class reviewed how to prove sets are equal using A={ 2, -2} and B = {x  : x²  = 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 x²  = 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, x  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 = A^c intersect B^c. We will discuss the proof of this next week.
 

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 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:

Next time: Arguments.