Date
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What we discussed/How we spent our time
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Aug 24
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Syllabus. Text. Why
does 2+2=4? Set Theory: Informal definition of a set.
List of axioms.
Notation for
sets. Axioms of Extentionality, Empty Set and Union. Successor
operation. Definitions of 0, 1, 2.
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Aug 26
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Definition of inductive set.
Axiom of Infinity.
Definition of the natural numbers as the least inductive set. Recursive
definitions of addition, multiplication and exponentiation. Informal definition
of "theorem" and "proof". Proof of the theorem "2+2=4".
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Aug 28
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Axiom of Pairing.
Definition of subset. Axiom of Power Set. Axiom of Comprehension.
Russell's Paradox. Axiom of Foundation.
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Aug 31
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Axiom of
Replacement. Axiom of Choice. Intersection of sets.
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Sep 2
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Proof that the intersection of all inductive sets is an inductive set
(= N). Derivation of the Principle of Mathematical Induction.
Examples of proofs by induction.
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Sep 4
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Variations on proof by induction, including Strong Induction.
Practice problems.
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Sep 9
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Use of induction to establish arithmetical laws.
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Sep 11
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First lecture on logic.
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Sep 14
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We discussed truth tables, tautologies and contradictions,
equivalent propositions. We learned why every proposition
is equivalent to one expressed with only "and" and "not".
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Sep 16
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We learned that every proposition is equivalent to
a proposition in disjunctive normal form, and we learned
a procedure for putting a proposition in disjunctive normal form.
We showed that
(H ⇒ C), ((¬C) ⇒ (¬H)), and
((H ∧ (¬ C)) ⇒ false) are equivalent, but are not
equivalent to (C &rArr H). We proved:
Thm. If 0 < x < 1, then x2 < x.
in three ways: we gave a
direct proof, we proved the contrapositive, and we proved the
theorem by contradiction.
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Sep 18
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We learned the definitions of "operation",
"predicate", "structure", "term",
"atomic formula" and "formula".
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Sep 21
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We learned about term trees and formula trees.
We worked in groups to practice writing formal sentences.
Solutions to practice problems.
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Sep 23
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We discussed how to determine if a quantified statement is true
in a structure.
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Sep 25
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Practice problems about quantifiers.
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Sep 28
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Prenex normal form.
Practice problems.
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Sep 30
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Definition of "relation". Relations encode mathematical
concepts. Definition of "function". Representations
of functions. Composition of functions.
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Oct 2
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Definitions of injection, surjection, bijection
(with examples). Definitions of
"image", "kernel", "inclusion map", "natural map"
and "induced map". Definitions of "equivalence relation",
"equivalence class". Proved that
the kernel of a function is an equivalence relation.
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Oct 5
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Description of all equivalence relations on a 3-element set.
Definition of "partition".
Relationship between partitions and equivalence relations.
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Oct 7
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Review of equivalence relations and partitions.
Proof that every equivalence relation is the kernel of a function.
Description of the canonical factorization of a function:
F = (inclusion) o (induced bijection) o (natural map)
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Oct 9
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Order relations: strict and nonstrict orders.
Irreflexivity, asymmetry, antisymmetry.
Partially ordered sets (= posets) and totally ordered sets.
Relationship between strict and nonstrict orders.
Difference between "maximal" and "maximum".
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Oct 12
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Description of the poset of all partial orders
on sets of size 2 or 3. Extending an order. Linear extensions.
Proof that every maximal extension of a partial order is a linear order.
Proof that
every (finite) partial order is the intersection of its linear
extensions.
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Oct 14
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Review.
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Oct 16
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Exam.
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Oct 19
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The integers: review of some properties of the natural numbers,
including the definition of the standard order on N.
Proof that N is a well ordered set. Proofs that:
0 is not a successor, every nonzero natural number is a successor,
successor is a 1-1 function, addition of natural numbers is cancellative.
Definition of the set of integers as equivalence classes of pairs
of natural numbers.
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Oct 21
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The integers: Proof that the relation used in the definition
of the integers is really an equivalence relation. Definition
of the arithmetic on Z. Discussion of the meaning
of "well-defined". Proofs that the arithmetic
structure on Z is well-defined.
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Oct 23
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The integers: Proofs of some laws of arithmetic
for Z.
Proof that every
integer is of the form mZ or
-mZ for some m∈ N.
Definition of the order on Z.
Proof that the order is well defined.
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Oct 26
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The integers: Homomorphism, isomorphism.
The map m → mZ
is a 1-1 homomorphism of N into Z,
which allows us to think of the natural numbers as a substructure
of the integers.
Definition of "cardinality", |A|=|B|, |A|=m, "finite".
Sketch of a proof of the Sum Rule and the Product Rule.
Proof that there are nm functions from
an m-element set to an n-element set.
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Oct 28
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Canceled due to snow.
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Oct 30
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Counting: There are n! ways to linearly order
an n-element set.
The number of subsets of an
n-element set is 2n.
The number of characteristic functions on an
n-element set is 2n.
The number of k-element subsets of an n-element set
is C(n,k) = n!/(k! (n-k)!), which is read "n-choose-k".
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Nov 2
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Counting:
Two proofs of Pascal's Identity. Pascal's Triangle.
Proof of the Binomial Theorem. We worked out some
exercises involving the Binomial Theorem.
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Nov 4
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Counting ordered partitions. Multinomial coefficients.
Multinomial Theorem. Pascal's Identities for multinomial
coefficients. Pascal's Pyramid.
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Nov 6
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The Principle of Inclusion and Exclusion.
Formula for counting the number of surjective
functions from an n-element set to an m-element set.
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Nov 9
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More on the Principle of Inclusion and Exclusion.
Counting derangements. Practice problems.
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Nov 11
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Stirling numbers and Bell numbers: Definitions and notation.
Examples. Recursion for the Stirling numbers. Table of small
values.
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Nov 13
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Stirling numbers and Bell numbers: Formula for Stirling
numbers. Binomial-type Theorem for Stirling numbers.
We worked on practice problems.
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Nov 16
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Formulas for distributions.
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Nov 18
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Undergraduate Committee Survey.
Practice problems about distributions.
Solutions.
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Nov 20
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Discrete probability: sample space, sample point, simple and compound
events, probability distribution, uniform distribution.
Practice exercises.
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Nov 30
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Probability practice problems.
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Dec 2
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Graph theory: definition of graph, multigraph, directed graph,
adjacency, incidence. Examples: paths, cycles, complete graphs,
complete r-partite graphs. Definition of planar drawings
and planar graphs. Euler's Formula: v-e+r=2. (Proof
was by induction on edges using contraction of
edges in the inductive step.)
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Dec 4
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Description of the five Platonic solids. Explanation of why the
vertices and edges of any convex polyhedron describe
a planar graph. More generally, a graph is planar iff it
can be drawn on the sphere without edges crossing.
Proof that if G is connected, loopless, bipartite,
and has at least 3 vertices, then e ≤ 3v-6.
Proof that
K5
is not planar. Corollary: Any graph that
has a subgraph that is a subdivision of
K5 is nonplanar.
In particular,
Kn is nonplanar iff n ≥ 5.
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Dec 7
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Bipartite graphs. Valid colorings. k-colorability.
Proof that the following are equivalent:
(1) G is bipartite.
(2) G is 2-colorable.
(3) G has no odd cycles.
(4) The lengths of two paths connecting distinct vertices
u and v in G have the same parity.
Proof that if G is connected, loopless, bipartite,
and has at least 4 vertices, then e ≤ 2v-4.
Proof that
K3,3
is not planar.
Statement of Kuratowski's Theorem: a (finite) graph
is planar iff it contains no subgraph that is a subdivision
of
K5
or
K3,3.
Example: the Petersen graph has a subgraph that is a subdivision
of K3,3.
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Dec 9
|
The classification of compact 2-manifolds: Description of a
compact 2-manifold. In what sense will we consider two
2-manifolds
to be the same? The approach to classification via
cutting and gluing.
Using the Euler characteristic to distinguish
between 2-manifolds. Computing the Euler
characteristic of the n-hole torus.
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Dec 11
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Review for the final.
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