2
BRIDGE TO ABSTRACT MATHEMATICS – LECTURE NOTES
1.
Introduction
This set of lecture notes consists of four main sections, to be covered sequentially.
A brief overview of the focus for each section is given below.
The first type of proof, by
deduction
1
, uses direct reasoning starting from
given facts, or
premises
. The accepted rules by which we can establish new facts
starting from known facts are formalized under the name of
mathematical logic
.
However, before defining mathematical logic, we need basic language to express
mathematical statements, and this is the first reason to introduce elementary
set
theory
.
The second type of proof, known as
mathematical induction
, is an extension
of the obvious idea of “proof by verification”, where there are only a finite number
of concrete cases to consider. However, when the number of (verifiable) concrete
instances that the statement can take is not finite, this
algorithmic
approach is
not sufficient. Sometimes, the infinite number of instances can be
counted
, just
like we count with natural numbers.
Therefore, the problem of comparing sets
(especially, infinite sets) arises, and through it, the notion of
function
(regarded
as a way of comparing two sets).
Another notion which arises at this point is
that of
equivalence relations
, since two equinumerous sets are, in a certain sense,
equivalent. Finally, when all the instances “to verify” form a countable set, and we
have an algorithm that allows us to prove one instance using the previous ones, we
arrive at the concept of mathematical induction.
When the statement to prove does not reduce to a countable number of instances,
it seems at first that we have no way to proceed. Luckily, principles of mathematical
logic provide a direct way of proving some of these statements, through the notion of
contradiction
. No matter how complicated the statement might look like, we as-
sume that it is
true
, and proceed to deductively derive other consequences it should
have. If one of these consequences turns out to be
false
, however, then principles of
logic dictate that the only possible interpretation of the apparent
contradiction
is that the original assumption was, in fact, unwarranted, and therefore its logical
“opposite” (negation) must be true instead. Areas of mathematics where proofs by
contradiction are encountered frequently are number theory and
topology
, which is
the reason behind introducing some basic topological concepts in this section.