Switching gears just a bit, we are going to introduce a concept that is prevalent
throughout mathematics, namely the equivalence relation. As the name implies, these
are special relationships between members of a set, and are foundation for many more
comp
We have used the word set a few times before this point, but only in the loosest, most intuitive sense. Now its time to dene exactly what a set is, mention some
properties we impose upon sets, and proove theorems that will help us later. We will
also intr
(3.2) Inclusion Proofs
Now that we have laid the foundation for dealing with sets and subsets, it is time to
prove certain properties of sets. This chapter will deal with the relationship between
various subsets, and whether two seemingly dierent sets are
Weve spent the past two sections on dealing with sets. Although you will encounter sets and make use of them in many future courses, for the most part, we will
not need to use some of the more advanced set theoretic techniques. We did, however, need to es
(2.2) More Proofs
Last chapter, we worked on proofs that could be tackled directly. Of course, it
will not always be the case that we can use the assumptions to prove the result. For
statements that cannot be proven in this manner, we will attempt to use
(2.3) Mathematical Induction
Look at the following question:
What is the sum of the rst n consecutive positive odd numbers?
How would a student go about trying to gure this one out? When we ask for
the rst n odd positive odd numbers, what exactly do we me
What is a proof? We probably all have an intuitive understanding of the concept,
but lets state it anyway. A mathematical proof is a way to show that if an assumption is true, then a resulting statement is true. If something is proven, there can
be no dou
In the previous two sections, we started to build a language by which we will interpret future problems and theorems. In this section, we dene the concept of an
operator (even though we have been using them in these notes) and introduce some
less common o
(1.1) The Language of Math
Right now, hopefully, you are reading this set of notes. So far it hasnt been very
heavy- a chapter title and two sentences separated by a period. There was a set of
parenthases above to note the chapter number, and a dash to se
(1.2) Quantiers
In the previous chapter, we covered a few of the symbols that represent dierent
sets, and some simple notation pertaining to proof statements. Now we will expand
this language to include something in math called a QUANTIFIER. A quantier is