# Math 74 Schedule

Here is an outline/summary of what we will cover/what we have covered on each
day.

This schedule serves at least three purposes. First, it gives me an idea of how
to pace the

class in order to cover all the material, and second, it allows the students to
estimate what

will be covered to prepare ahead of time, and third, if a student must miss
class, it will allow

them to see what they have missed.

**1 Statements, Implications, and Logical Connectives **

Class 1 (August 28).

introduction

statements

truth value

truth tables

implications (P -> Q)

inverse, converse, and contrapositive

the logical connectives “and”

## 2 Sets

Class 2 (September 2).

sets, elements of sets (∈)

empty set (Φ)

containment
,
subsets, equality of sets

operations on sets : union
,
intersection
,
complement ( difference ) (X \ Y ,
for

),
(Cartesian) product (X × Y )

disjoint sets ()

proper subsets

power set (P(X)).

Class 3 (September 4).

quantifiers: for all
there exists

negation of quantifiers

statements that rely on sets, proving statements that rely on sets

the following statements are equivalent :

## 3 Relations

Class 4 (September 9).

relations (R)

reflexive, symmetric, antisymmetric, transitive, and skew-transitive relations

equivalence relations

partial orders

## 4 Induction, Strong Induction, and the Well Ordering Principle

Class 5 (September 11).

statements depending on

induction

strong induction

Class 6 (September 16).

the well ordering principle

prime numbers

the equivalence of induction, strong induction, and the well ordering principle

## 5 Functions

Class 7 (September 18).

domain, codomain, rule definition of a function

image

injective, surjective, and bijective functions

graph

graph defintion of a function

equality of functions

composition of functions, associativity of function composition

Class 8 (September 23). Review for Exam 1.

Class 9 (September 25). Exam 1.

Class 10 (September 30).

image and preimage

restriction and corestriction of functions

## 6 Invertible Functions

Class 11 (October 2).

invertible functions

left, right, and two sided inverses

theorem relating (left, right, two sided) inverses with (injective, surjective,
bijective) functions

axiom of choice

the set [n] for

the injections
and the surjections

## 7 Cardinality

Class 12 (October 7).

the pigeonhole principle

finite sets, cardinality (#X = n)

infinite sets

countable sets

denumerable sets, cardinality

uncountable sets

Class 13 (October 9).

the subset/cardinality lemma

a set is countable <-> it is finite or denumerable

reconciling two definitions of cardinality (for countable sets) (#X and |X|).

Class 14 (October 14).

the product of two denumerable sets is denumerable

Q is countable

R is uncountable

sets of functions

cardinality of such sets

relatively prime integers

## 8 Groups

Class 15 (October 16).

binary operations

magmas: associative, commutative, (left, right) cancellative

(left, right) identities and inverses

unital magmas, semigroups, monoids, (commutative) groups

Class 16 (October 21).

the group S _{n}

equivalence classes

congruence modulo

the group Z/n

more examples of groups

Class 17 (October 23).

more examples of groups

subgroups

group homomorphisms

definition of a ring, field

definition of a ring homomorphism

Class 18 (October 28). Review for Exam 2.

Class 19 (October 30). Exam 2.

## 9 Examples of Rings and Fields

Class 20 (November 4).

definition of a ring R and a field k

the rings Z, Z/n, M_{n}(R)

Class 21 (November 6).

the set of divisors D (n)

greatest common divisors (gcd(m, n))

the division theorem

the set of all primes is denumerable

Z/n is a field if and only if n is prime.

## 10 Metric Spaces

Class 22 (November 13).

metrics (distance)

metric spaces (X, d)

open ball at x of radius r (B_{r}(x))

open, closed sets

Class 23 (November 18).

arbitrary union [intersection] of open [closed] sets is open [closed]

finite intersection [union] of open [closed] sets is open [closed]

give the same open sets

Class 24 (November 20).

sequences (x_{n})

convergent and Cauchy sequences

convergent implies Cauchy

completeness

limit points, set of limit points

Class 25 (November 25).

a set S is closed if and only if

every convergent sequence is bounded

three definitions of a continuous function:

3. preimage of open sets are open

## 11 Complex Numbers

Class 26 (December 2).

complex numbers

the unit circle (S^{1})

S^{1} is a group

series

Euler’ s Formula

polar coordinates

Class 27 (December 4).

Class 28 (December 9).

Review for Final Exam.

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