Math 74 Schedule
Here is an outline/summary of what we will cover/what we have covered on each
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).
implications (P -> Q)
inverse, converse, and contrapositive
the logical connectives “and”
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 ()
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 :
Class 4 (September 9).
reflexive, symmetric, antisymmetric, transitive, and skew-transitive relations
4 Induction, Strong Induction, and the Well Ordering Principle
Class 5 (September 11).
statements depending on
Class 6 (September 16).
the well ordering principle
the equivalence of induction, strong induction, and the well ordering principle
Class 7 (September 18).
domain, codomain, rule definition of a function
injective, surjective, and bijective functions
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).
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
Class 12 (October 7).
the pigeonhole principle
finite sets, cardinality (#X = n)
denumerable sets, cardinality
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
Class 15 (October 16).
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
the group Z/n
more examples of groups
Class 17 (October 23).
more examples of groups
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, Mn(R)
10 Metric Spaces
Class 22 (November 13).
metric spaces (X, d)
open ball at x of radius r (Br(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).
convergent and Cauchy sequences
convergent implies Cauchy
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
Class 27 (December 4).
Class 28 (December 9).
Review for Final Exam.