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
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 pro
(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
vari
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 advance
(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 c
(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
th
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 stateme
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 usin
(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
parenth
(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 inclu