MATH 240 Introduction to Mathematical Thought
Fall, 2006      Class Summaries

Introduction to Sets
Representation of Set
Basics(Equality, Subset, Etc)
Optional on-line exercises 1
Set Operations
Optional on-line exercises 2
Properties of Set Operation

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.


The nature of the subset relation. Focus 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.

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.
 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".
Observe 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 ask "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.