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Math 2001-001: Intro to Discrete Math, Spring 2019
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Lecture Topics
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Date
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What we discussed/How we spent our time
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Jan 14
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Syllabus. Policies. Text.
The main goals of the course are defined to be:
(1) To learn what it means to say
``Mathematics is constructed to be well founded.'' To learn
which concepts and assertions depend on which others.
To learn what are the most primitive
concepts ( = set, $\in$) and the most primitive
assertions ( = axioms of set theory).
(2) To learn how to unravel the definitions of
``function'', ``number'', and ``finite'',
through layers of more and more primitive
concepts, back to ``set'' and ``$\in$''.
(3) To learn the meanings of ``truth'' and ``provability''.
To learn proof strategies.
(4) To learn formulas for counting.
(5) (If time) To learn elements of graph theory.
Axioms of set theory.
I will occasionally post notes for Math 2001
in the form of flash cards on Quizlet. To join
our quizlet class, go
https://quizlet.com/join/mExWGGZqj.
(Test yourself on the Axioms of Set Theory with this
Quizlet link:
https://quizlet.com/_61ko6h.)
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Jan 16
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We began discussing naive set theory alongside axiomatic set theory.
We discussed the Axiom of Extensionality and the Axiom of the Empty Set.
We introduced Venn diagrams, the directed graph representation of the
universe of sets, and the concept of the successor of a set.
(Some Venn diagrams from the web:
1,
2,
3. The second one is not mathematically correct.)
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Jan 18
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We discussed the Axioms of Infinity, Pairing, and Union.
We defined the natural numbers to be the intersection
of all inductive sets.
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Jan 23
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First we defined subset and power set,
and introduced the Axiom of Power Set.
Then we turned to a discussion of abbreviations
in mathematics. We defined the alphabet for set theory
(variables, nonlogical symbols, logical symbols, punctuation),
and how to write formal definitions for predicates.
Worksheet 1 (+ solution sketches).
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Jan 25
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We discussed the relationship between
unrestricted comprehension and restricted
comprehension. We showed, through
Russell's Paradox, that the rule of
unrestricted comprehension leads to a contradiction.
We derived that there is no set of all sets.
We also explained why there is no set
containing all sets except one.
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Jan 28
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We wrote the proof of the theorem
``$R=\{x\;|\;x\notin x\}$ is not a set''
in English. Then we discussed some of the history of the axioms of
set theory, including the introduction of the Axiom of Replacement.
(Because of the bad weather, and the fact that 30 percent of the class
was absent, the Monday quiz was made into an
ungraded practice worksheet.)
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Jan 30
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Read pages 20-27.
We discussed the Axiom of Choice and the Axiom
of Regularity/Foundation. In the discussion
of the Axiom of Foundation we defined the notion
of an $\in$-minimal element of a set.
We defined ZFC (all 10 axioms) and
ZF (all 10 axioms minus the Axiom of Choice).
We discussed classes (like the class of all sets),
explaining what classes are
and how they may differ from sets.
We showed that the Axiom of Separation
allows us to intersect any nonempty
class of sets.
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Feb 1
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Today we discussed De Morgan's Laws,
$\overline{X\cap Y} = \overline{X} \cup \overline{Y}$ and
$\overline{X\cup Y} = \overline{X} \cap \overline{Y}$.
We also noted that
$X\subseteq Y$ if and only if
$\overline{X}\supseteq \overline{Y}$.
Here $\overline{X}$ represents the complement of $X$
relative to some large set $A$. By referencing De Morgan's laws,
we reasoned that $\cup$ and $\cap$ satisfy ``dual'' properties.
As an example of this duality,
we proved $X\subseteq X\cup Y$ directly, and then
derived from this,
by duality, that $X\supseteq X\cap Y$.
(Quizlet allows me to write $X'$ but not $\overline{X}$,
so on Quizlet I will write De Morgan's Laws as
$(X\cap Y)' = X' \cup Y'$ and
$(X\cup Y)' = X' \cap Y'$.
Despite this duality,
we noted that there are some asymmetries between
union and intersection. The first asymmetry we noted was
that $\bigcup \emptyset$ is a valid set (it is $\emptyset$),
but $\bigcap \emptyset$ is not set.
The second asymmetry we noted about
union and intersection is that we can only
form the union of a collection that is a set, but we can
form the intersection of any collection that is a class.
It is important that we can form these types
of intersections, since the set of natural numbers,
$\mathbb N$, is defined as the intersection of the class
of inductive sets. (We explained today why the class of
inductive sets is not a set.)
Finally, we introduced the Kuratowski encoding of ordered pairs,
namely $(a,b) := \{\{a\},\{a,b\}\}$. We stated that,
with this definition the following theorem holds:
Theorem. $(a,b)=(c,d)$ if and only if
$a=c$ and $b=d$. (Not proved yet!)
(Test yourself on set theory terminology with this
Quizlet link:
https://quizlet.com/_61ufo1.
Some of these definitions are illustrated
by examples here
https://quizlet.com/_61vmmh.)
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Feb 4
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We proved
Theorem. $(a,b)=(c,d)$ if and only if
$a=c$ and $b=d$.
We defined ordered triple, ordered $n$-tuple, and
Cartesian product $A\times B$. We explained why,
if $A$ and $B$ are sets, then $A\times B$ is also a set.
Quiz 1.
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Feb 6
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Read pages 35-47.
We defined relations and explained the connection
between relations and predicates.
Handout on relations.
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Feb 8
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Read pages 28-35.
We defined functions and went over some
vocabulary for functions.
Quiz yourself!
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Feb 11
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Read pages 47-51.
What type of mathematical object is a
domain, codomain, image, or coimage?
Any domain is a set and any set is a domain.
Any codomain is a set and any set is a codomain.
We defined the identity function on $S$, ${\rm id}_S:S\to S$,
to show how to realize any set $S$ as a domain or a codomain.
The image of a function is a subset of the codomain.
Conversely, if $B$ is any set and
$S\subseteq B$ is any subset, then
there is a function, $\iota_S:S\to B$,
the inclusion function for $S$ into $B$, for which $B$ is the codomain
and $S$ is the image.
The coimage of a function is a partition of the domain.
Conversely, if $A$ is any set and
$P$ is any partition of $A$, then
there is a function, $N_P:A\to P: a\mapsto [a]$,
the natural map for $P$, for which $A$ is the domain
and $P$ is the coimage.
Quiz 2.
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Feb 13
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Read pages 51-65.
Directed graph representation of binary relations.
Equivalence relations are the abstraction of kernels.
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Feb 15
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Recursion and induction, I.
Exploiting the defining property of $\mathbb N$.
Why is recursion a valid way to define a function?
Why is induction a valid form of proof?
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Feb 18
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Recursion and induction, II.
Proving the laws of arithmetic.
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Feb 20
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More induction proofs.
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Feb 22
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Cardinality, I. Finite versus infinite.
Countable versus uncountable.
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Feb 25
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Cantor's Theorem.
Cantor-Schroeder-Bernstein Theorem.
$|\mathbb N|<|{\mathcal P}(\mathbb N)|=|\mathbb R|=|\mathbb R^n|$.
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Feb 27
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Review for the midterm!
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Mar 1
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Midterm!
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Mar 4
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Structures. Sentences about structures.
Truth.
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