Date
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What we discussed/How we spent our time
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Jan 15
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Syllabus. Policies. Text.
I define the main goals of the course 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, and the distinction between,
``truth'' and ``provability''.
To learn proof strategies.
(4) To learn formulas for counting.
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 20
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Read Sections 1.1 and 2.1.
We discussed naive set theory
following these slides.
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Jan 22
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We discussed five of the first six axioms of set theory
following these slides.
On the last page of these slides we discuss
the use of definitions/abbreviations
to introduce new symbols into mathematics (like the symbols
$\emptyset, \subseteq, {\mathcal P}$.)
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Jan 25
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We discussed the
Axiom of Separation
(also called the Axiom of Restricted Comprehension).
We also discussed the use of the symbols
$\cup$ and $\bigcup$.
During the last 10 minutes we took Quiz 0.
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Jan 27
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We discussed
Union and Intersection,
noting that the union of the empty collection
is defined while the intersection of the empty collection
is not defined. We also noted that sets
$X$ and $Y$ are called ``disjoint'' if $X\cap Y=\emptyset$.
We introduced the Axiom of Infinity,
and used it to show that
the intersection of all inductive sets,
which we denote $\mathbb N$,
is a set.
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Jan 29
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We discussed
Russell's Paradox.
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Feb 1
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We discussed the construction of
ordered pairs, triples and $n$-tuples.
We showed that ordered pairs have the property that
$(a,b)=(c,d)$ holds if and only if $a=c$ and $b=d$.
We defined the Cartesian product of sets, and showed
that, if $A$ and $B$ are set, then $A\times B$ is a set.
We took Quiz 1.
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Feb 3
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We discussed the results of Quiz 1.
We ended with a Q&A session (extended office hours).
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Feb 5
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Read Sections 2.2 and 2.3.
We discussed relations, predicates, and functions.
Quiz 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.
Quiz yourself
on terminology for functions!
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Feb 8
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We started by working on
this worksheet
concerning functions.
Then we began discussing
these slides
on functions. These slides refer to some terminology
for functions. The terminology is defined fully on
this handout.
We took Quiz 2.
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Feb 10
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We continued discussing
these slides,
focusing on partitions. We showed that the
coimage of any function is a partition and
that every partition is a coimage.
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Feb 12
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We defined ``kernel of a function'',
and abstracted the kernel concept
to obtain the definition of equivalence relation.
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Feb 15
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Read Section 2.5.
We discussed the proof that
the kernel of any function is an equivalence relation on the domain
of the function,
and every equivalence relation on a set $A$
is the kernel of some function with domain $A$.
This proof introduced the concept of ``equivalence class''.
We took Quiz 3.
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Feb 17
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Wellness Day!
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Feb 19
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Midterm Review Sheet!
We completed the proof that the kernel of a function
is an equivalence relation, and every equivalence
relation is the kernel of a function.
We reviewed the concept of ``well foundedness''
and how the definition of a ``function''
is developed from the axioms.
We began discussing induction, recursion,
and the arithmetic of $\mathbb N$.
(Some hints.)
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Feb 22
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Midterm Review Sheet!
We continued discussing induction, recursion,
and the arithmetic of $\mathbb N$.
(Some hints.)
We took Quiz 4.
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Feb 24
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Midterm Review Sheet!
We reviewed for the midterm exam.
(No new HW assigned for the week of Feb 24-March 3!)
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Feb 26
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Midterm!
Answers.
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Mar 1
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No quiz today!
Instead we discussed the
last three axioms of set theory.
(Replacement, Choice, Foundation)
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Mar 3
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No HW due today!
We started proving the
laws of arithmetic.
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Mar 5
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We proved the Distributive Law by induction.
We defined the concepts:
finite, infinite, countably infinite, countable,
uncountable.
We proved the a subset of a finite set is
finite by induction.
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Mar 8
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We defined the ordering on natural numbers:
$m\lt n$ means $m\in n$; $m\leq n$ means
$m\lt n$ or $m=n$.
Under these definitions, every natural number
is the set of its predecessors under the $\lt$-ordering.
We stated a theorem:
$m\leq n$ iff $m\subseteq n$.
We discussed Strong Induction.
We showed that it is a valid form of proof,
and used it to prove that every natural number $k\neq 0$
is a product of prime numbers. This fact
(existence of prime factorizations)
is half of the
Fundamental Theorem of Arithmetic. We stated the other
half (uniqueness of prime factorizations),
but did not prove that half.
We took Quiz 5.
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Mar 10
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Read: Sections 2.7 and 2.8.
We started discussing
ordinal and cardinal numbers.
We defined ordinal numbers and introduced some new symbols
($\omega$ = omega, $\aleph_0$ = aleph zero).
We discussed the main properties of ordinals,
including that ordinals are well ordered
and that every set can be enumerated by an ordinal.
We showed that $\omega$ can be enumerated
by each of $\omega, \omega+1, \omega+2$ and $\omega+\omega$.
We then defined equipotence, finite, infinite,
countably infinite, countable, and uncountable.
We introduced the notation
$|A|=|B|$, $|A|\leq |B|$, and $|A|<|B|$.
We started the proof of the CBS Theorem, but did not finish it.
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Mar 12
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We completed the proof of the CBS theorem.
We explained why $A\subseteq B$ implies $|A|\leq |B|$.
We showed that
$|\mathbb N|=|E|$ ($E$=even natural numbers),
$|\mathbb N|=|\mathbb N\times\mathbb N|$,
$|\mathbb N|=|\mathbb Q^{\geq 0}|$, and
$|\mathbb N|=|\mathbb Q|$.
We then explained why
$|(0,1)|=|[0,1]|=|\mathbb R|$,
and noted that these arguments can show that
if $S\subseteq \mathbb R$ is any subset of
$\mathbb R$ with nonempty interior, then
$|S|=|\mathbb R|$. (Saying that $S$
has nonempty interior means some nonempty
open interval is a subset of $S$.)
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Mar 15
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Snow day! (University is closed.)
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Mar 17
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We proved
Cantor's Theorem,
introduced
characteristic functions, and explained why
$|\mathbb N|< |{\mathcal P}(\mathbb N)|=|\mathbb R|$.
We took Quiz 6.
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Mar 19
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We discussed Quiz 6.
Review: Sections 2.7 and 2.8.
We completed
these notes,
in particular we discussed why (in ZFC)
every set has a unique cardinality.
We also discussed the Continuum Hypothesis.
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Mar 22
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We are starting to study logic (Chapters 3 and 4).
In Chapter 3, we will discuss material from
Sections 3.1 and 3.5,
and part of 3.2. We will not discuss material from
3.3, 3.4, or 3.6.
Two introductory handouts
on logic.
No quiz!
Arnie has no respect for those who have no respect for logic.
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Mar 24
|
Read Subsection 4.1.3.
We talked about structures and their structural
elements. We talked about tables for operations and predicates.
(See notes from Feb 5 lecture, where operations and
predicates were defined for the first time.)
Practice problems on creating
tables for compound operations and predicates.
Solutions.
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Mar 26
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We discussed formula trees and
composition trees/parse trees/derivation trees/syntax trees.
We gave the recursive definition for terms
and explained how to assign tables to
terms. We gave the truth tables for the
Boolean logical connectives
$\wedge, \vee, \neg, \to, \leftrightarrow$.
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Mar 29
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We discussed
tautologies, contradictions,
logical equivalence.
We took Quiz 7.
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Mar 31
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We continued discussing these slides,
and explained why every proposition is equivalent
to one in disjunctive normal form.
We then started to discuss quantifiers,
and got far enough to explain why
$\neg (\forall x) P\equiv (\exists x) \neg P$
and $\neg (\exists x) P\equiv (\forall x) \neg P$.
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Apr 2
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We discussed the rules to put a formula in
prenex normal form.
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Apr 5
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We discussed how to standardize
the variables apart, which helps
to put a statement in prenex form.
We then began discussing quantifer games for sentences
in prenex form. We argued that $\exists$ has a winning strategy
for $(\exists y)(\exists x)((x<0)\wedge (y>0))$
in $\mathbb R$, while $\forall$ has a winning
strategy for the same sentence in $\mathbb N$.
I ended the lecture by asking who has a winning strategy
for $(\forall x)(\exists y)(x=y)$ in $\mathbb R$?
We took Quiz 8.
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Apr 7
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We discussed how to interpret
quantifiers in general,
and more specifically how to use quantifier games
to determine
the truth of a sentence written
in prenex normal form.
Practice problems.
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Apr 9
|
We discussed provability versus truth.
The discussion included
(1) Semantic versus syntactic consequence. ($\Sigma\models P$ versus
$\Sigma\vdash P$.)
(2) The definition of ``proof''.
(3) The role of axioms.
(4) Examples of rules of deduction (e.g. ``Modus Ponens'').
(5) Soundness and completeness of a proof system.
(6) First-order sentences.
(7) Gödel's Completeness Theorem.
(8) The structure of a theorem statement.
(9) Proof strategies.
(10) The equivalence of direct proof, proof of the contrapositive,
and proof by contradiction.
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Apr 12
|
We completed these slides.
The main new item was a discussion
of proof when the hypotheses or conclusions
involve quantifiers.
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Apr 14
|
Read Section 6.1.
We discussed the first page of
this handout.
In particular, we discussed the additive and multiplicative
counting principles.
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Apr 16
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We discussed more of
this handout.
In particular, we argued that
(1) an $n$-element set has $2^n$ subsets,
(2) the number of functions from a $k$-element set
to an $n$-element set is $n^k$,
(3) the number of bijective functions from a $k$-element set
to an $n$-element set is $0$ is $k\neq n$
and is $n!$ if $k=n$,
(4) the number of injective functions from a $k$-element set
to an $n$-element set is $(n)_k=n!/k!$, and
(5) the number of $k$-element subsets
of an $n$-element set is $n!/(k!(n-k)!)$.
During this discussion, we discussed overcounting,
in the following form:
If $E$ is an equivalence relation on $L$, and all
$E$-classes have the same size $k$, then the number
of $E$-classes is $|L|/k$.
(Special case: You can count the number of cows in a field
by counting their legs and dividing by four.
Here $L$ is the set of legs, and two legs
are called equivalent if they belong to the same cow.
We assume that each cow has $k=4$ legs.)
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Apr 19
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We discussed ``$n$-choose-$k$''.
(1) The $k$th level of the subset lattice of an $n$-element set
has size $C(n,k)$.
(2) Formulas for $C(n,k)$ are $(n)_k/k!=n!/k!(n-k)!$
(3) The number of poker hands is $C(52,5)=2,598,960$.
The number of poker hands that are Royal Flushes is $C(4,1)=4$.
The probability that a poker hand is a Royal Flush is calculated by the ratio
$(\# \textrm{successful events})/(\# \textrm{events})$,
which in the case of a Royal Flush is
$C(4,1)/C(52,5)=.000001539$.
(4)Pascal's Triangle.
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Apr 21
|
We discussed
binomial coefficients,
multinomial coefficients, and
the multichoose numbers.
We have now covered all the counting formulas in
this handout
except for the formula involving $S(k,n)$.
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Apr 23
|
Final Review Sheet.
We discussed
inclusion and exclusion,
the Stirling numbers
of the second kind, and the number of surjective
functions $f\colon k\to n$.
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Apr 26
|
We discussed
the Bell numbers and
Stirling numbers of the second kind.
$B_n$ counts the number of partitions
of an $n$-element set, and $S(n,k)$ counts the number of
partitions of an $n$-element set into $k$-cells.
Quiz yourself on
counting formulas!
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