Date
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What we discussed/How we spent our time
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Aug 23
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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 ``infinite'',
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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Aug 25
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Read Sections 1.1 and 2.1.
We began discussing `naive' set theory,
in which we `define' a set to be
an unordered collection of distinct objects.
We contrasted this with formal set theory
based on the axiom system ZFC (Zermelo-Fraenkel
set theory with the Axiom of Choice).
We described the directed graph model of ZFC. We
introduced
(i) the symbol $\in$.
(ii) the Axiom of the Empty Set.
(iii) the Axiom of Extensionality.
(iv) the successor function $S(x)=x\cup \{x\}$.
(v) the definitions of $0, 1, 2, 3, 4$.
(vi) the recursive definition of addition of natural numbers.
Using these definitions, we proved that $2+2=4$.
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Aug 27
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We reviewed naive set theory versus formal set theory.
We discussed four of the first five axioms of set theory
(Extensionality, Empty Set, Pairing, Union).
We discussed
the use of definitions/abbreviations
to introduce new symbols into mathematics (like the symbols
$\emptyset, \bigcup, S(x)$).
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Aug 30
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We defined inductive set.
We introduced the Axiom of Infinity
and the Axiom of Restricted Comprehension.
We defined $\mathbb N$
to be the set of elements common to all
inductive sets.
We proved that $\mathbb N$ is an inductive set,
and noted that it is the intersection of
all inductive sets, and therefore is
the least inductive set.
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Sep 1
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We introduced the concept of a subset of a set,
defined the notation $x\subseteq y$,
and stated the Power Set Axiom.
We proved that $A=B$ if and only if $A\subseteq B$
and $B\subseteq A$.
We then proved the distributive law
$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$.
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Sep 3
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We discussed the following questions.
(i) What is naive set theory?
(ii) Why is naive set theory inconsistent?
(iii) Why should we have expected naive set theory to be inconsistent?
(iv) Do the axioms of ZFC protect us from contradictions?
(v) What kind of object is $\{x\;|\;x\notin x\}$? Is it still possible
to discuss such objects mathematically?
Our discussion involved defining the
Russell class,
a discussion of Russell's Paradox,
Gödel's Second Incompleteness Theorem, and
classes versus sets.
We spent the last 15 minutes of class discussing
the statements and meanings of the Axioms of
Replacement, Choice, and Foundation.
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Sep 8
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We introduced the Kuratowski encoding
of ordered pairs: $(a,b)=\{\{a\},\{a,b\}\}$.
(i) We explained why $(a,b)$ is a set.
(ii) We proved that $(a,b)=(c,d)$ implies $a=c$ and $b=d$.
(iii) We defined ordered triples and quadruples
(iv) We defined the Cartesian product
$$A\times B = \{x\in{\mathcal P}{\mathcal P}(A\cup B)\;|\;x=(a,b), a\in A, b\in B\}$$
of sets $A$ and $B$.
We use the product symbol $\times$ for Cartesian products
when using infix notation,
$A_1\times A_2\times A_3\times A_4$, and the symbol $\prod$
when using prefix notation,
$\prod_{i=1}^4 A_i$. If all the factors are the same, we might
write $A^4$ for $A\times A\times A\times A$.
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Sep 10
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Read Sections 2.2 and 2.3.
We discussed how ordered pairs and $n$-tuples
are used to define relations and functions.
We discussed the notation $$f\colon A\to B\colon x\mapsto x^2.$$
We discussed the following
terminology about functions:
(1) Directed graph representation of a function.
(2) Domain of a function.
(3) Codomain of a function.
(4) Image of a function.
(5) Preimage of an element under a function.
(6) Fiber over an element.
(7) Coimage of a function.
(8) Inclusion function. (Or inclusion map.)
(9) Natural function. (Or natural map.)
(10) Induced function. (Or induced map.)
(11) Canonical factorization of a function.
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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Sep 13
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We reviewed all of the function terminology,
and saw how to write each concept down
for the squaring function
$$F\colon \mathbb R\to \mathbb R\colon x\mapsto x^2.$$
In addition, we defined
(1) Composition of functions.
(2) Injective function.
(3) Surjective function.
(4) Bijective function.
(5) Identity function.
(6) Inverse of a function.
Quiz 1 is at 5-7pm on Canvas!
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Sep 15
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We explained why the natural map is surjective, but
usually not injective.
We explained why the inclusion map is injective, but
usually not surjective.
We explained why the induced map is bijective.
We defined kernel, partition, and equivalence relation,
and explained the relationship coimages$\leftrightarrow$partitions,
namely the coimage of a function with domain $A$
is a partition of $A$, while any partition of $A$
is the coimage of a function with domain $A$.
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Sep 17
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Read Section 2.5.
We began to explain the relationship
kernels$\leftrightarrow$equivalence relations,
namely the kernel of a function with domain $A$
is an equivalence relation on $A$, while any
equivalence relation on $A$ is the
kernel of a function with domain $A$.
In this explanation, we introduced the concept
of equivalence class. Here,
if $E$ is an equivalence relation on $A$,
the set $[a]_E=\{x\in A\;|\;(a,x)\in E\}$
is the $E$-equivalence class of $a$.
The set $A/E=\{[a]_E\;|\;a\in A\}$ is a partition
of $A$ into $E$-equivalence classes.
Quiz yourself on coimages, kernels, partitions and equivalence relations
with this
Quizlet link:
https://quizlet.com/371093838/
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Sep 20
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We reviewed the concept of ``well foundedness'' and
how the definitions of a ``function'', ``number'', and ``finite''/``infinite''
are developed from the axioms.
We began discussing induction, recursion,
and the arithmetic of $\mathbb N$.
(Some hints.)
Quiz 2 is at 5-7pm on Canvas!
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Sep 22
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We discussed examples of definition
by recursion and proof by induction.
(This included formulas for the sum of the first
$n$ positive integers, the sum of the first
$n$ positive odd integers, the sum of the first $n$ squares,
the sum of the first $n$ cubes, and the formula for the
sum of a geometric series.)
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Sep 24
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We discussed other forms of recursion
(recursion using parameters and course-of-values
recursion).
Then we proved the laws of successor and the laws
of addition from the handout on
the arithmetic of $\mathbb N$.
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Sep 27
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We discussed
(1) Induction starting at $s_n$ instead of $s_0$.
(2) False induction proofs.
(3) Course-of-values induction.
The example proofs we discussed were
(1) A correct proof by ordinary induction
that $2^n\leq n!$ for $n\geq 4$.
(2) A false proof by ordinary induction that when you put $n$ points
on the boundary of a disk and cut the disk through
all pairs you obtain $2^{n-1}$ regions.
(3) A false proof by ordinary induction that all horses
have the same color.
(4) A correct proof by strong induction that every positive
number is a product of prime numbers.
Quiz 3 is at 5-7pm on Canvas!
Midterm Review Sheet!
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Sep 29
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Read: Sections 2.7 and 2.8.
We started discussing
ordinal and cardinal numbers.
We defined ordinal numbers and introduced
the symbol $\omega$ (omega) as a symbol for
the first infinite ordinal. $\omega=\mathbb N$.
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 $|A|<|B|$.
Quiz yourself on ordinals and cardinals with this
Quizlet link:
https://quizlet.com/375068176/
Midterm Review Sheet!
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Oct 1
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We continued discussing
ordinal and cardinal numbers.
We proved the Cantor-Bernstein-Schroeder Theorem.
We derived the Corollary that $A\subseteq B\subseteq C$
and $|A|=|C|$ together imply that $|A|=|B|=|C|$.
Hence equipotence classes of ordinals are convex.
We showed that $|\mathbb N|=|\mathbb N\times \mathbb N|$
in two ways, one way using the CBS theorem and one way not using it.
Midterm Review Sheet!
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Oct 4
|
We continued discussing
ordinal and cardinal numbers.
We proved
(1) If $A$ is an alphabet satisfying $|A|\leq |\mathbb N|$ and
each element of a set
$X$ can be described by a finite-length
sentence in the alphabet $A$,
then $|X|\leq |\mathbb N|$.
(2) $|\mathbb Z|=|\mathbb N|$. (Describe integers with alphabet $A=\{-\}\cup \mathbb N$.)
(3) $|\mathbb Q|=|\mathbb N|$. (Describe rational numbers with alphabet $A=\{-, /\}\cup \mathbb N$.)
(4) $|\mathbb R|\leq |(0,1)|\leq |{\mathcal P}(\mathbb N)|\leq |\mathbb R|$.
(5) Cantor's Theorem.
(6) $|\mathbb N|\lt |{\mathcal P}(\mathbb N)|=|\mathbb R|$.
We also mentioned that $\omega_0$ is notation
for the first countable ordinal/cardinal, and
$\omega_1$ is notation
for the first uncountable ordinal/cardinal.
Quiz 4 is at 5-7pm on Canvas!
Midterm Review Sheet!
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Oct 6
|
We reviewed for the midterm. (The midterm will be
in-class on Friday October 8.)
Midterm Review Sheet!
Some hints!
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Oct 8
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Midterm!
Solutions!
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Oct 11
|
Read: pages 89-95.
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.
Today we discussed two introductory handouts
on logic. We introduced
the notions of proposition (= ``declarative sentence'', or
``sentence which can be assigned a truth value''),
propositional connective, and the
truth tables for $\wedge, \vee, \neg, \to, \leftrightarrow$.
No quiz!
Arnie has no respect for those who have no respect for logic.
Mathematical symbols.
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Oct 13
|
No HW due today! (New assignment posted for next week!)
Today we discussed propositional logic following
these slides.
(We got to page 6.)
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Oct 15
|
Today we completed
these slides.
Truth table review and practice!
Hints!
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Oct 18
|
Read Subsection 4.1.3.
We discussed more propositional logic, including
(1) the equivalences $H\to C\equiv (\neg C)\to (\neg H)\equiv (H\wedge (\neg C))\to \textrm{False}$.
(2) the contrapositive, converse, and inverse of $H\to C$.
(3) logical consequence and logical independence.
(We say that $Y$ is a logical consequence of $X$ if
$X\to Y$ is a tautology. $X$ and $Y$ are logically independent
if neither is a logical consequence of the other.)
We talked about structures and their structural
elements.
Quiz 5 is at 3-10pm on Canvas!
(This is a quiz on Propositional Logic. See the lecture
notes from October 11-15.)
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Oct 20
|
The due date for HW6 was moved to October 27.
Today we defined Terms, Atomic Formulas, Formulas,
We drew a Term Tree and a Formula tree.
Example: in the formula
$$
(\forall a)(\exists b)(\forall c)(\exists d)((a^2+b^2=c^2+d^2)\vee (a^2+c^2=b^2+d^2))
$$
the atomic formulas are
$F=(a^2+b^2=c^2+d^2)$
and $G=(a^2+c^2=b^2+d^2)$.
The expressions $a^2+b^2,c^2+d^2,a^2+c^2$, and $b^2+d^2$
are examples of terms. The formula has the logical structure
$$
(\forall a)(\exists b)(\forall c)(\exists d)(F\vee G)
$$
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Oct 22
|
The due date for HW6 has been moved to October 27.
Today we discussed tables for structural elements.
We worked on these
practice problems. Solutions!
We discussed the first 1.5 pages of
this
handout on quantifiers.
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Oct 25
|
The due date for HW6 has been moved to October 27.
Today we completed
this
handout on quantifiers. (We solved Practice Problem 5 in class.)
Quiz 6 is at 3-10pm on Canvas!
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Oct 27
|
We discussed
(1) how the meaning of a sentence
in a structure is determined by the tables
of the structural elements. (This was review from Oct 18-22!)
(2) how one can use quantifier games to determine truth or falsity.
(This was review from Oct 25!)
(3) how to put a sentence in prenex form.
(This was new material!)
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Oct 29
|
We finished discussing the prenex form handout.
In particular, we discussed how to standardize
the variables apart in a sentence and then to put
the sentence in prenex form.
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Nov 1
|
We reviewed and
practiced
what it means for a sentence to be
true in a structure. Then we discussed
semantic consequence. ($\Sigma \models P$.)
We discussed provability versus truth.
The discussion included
(1) The definition of ``proof''.
(2) Semantic versus syntactic consequence. ($\Sigma\models P$ versus
$\Sigma\vdash P$.)
(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.
Quiz 7 is at 3-10pm on Canvas!
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Nov 3
|
We discussed
(1) The structure of a theorem statement.
(2) Proof strategies.
(3) The equivalence of direct proof, proof of the contrapositive,
and proof by contradiction.
(4) Proofs involving quantifiers.
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Nov 5
|
We discussed Formal versus informal proofs.
Then we began a discussion of counting.
We stated the Sum Rule and the Product Rule,
and proved the Sum Rule.
Read Section 6.1.
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Nov 8
|
We discussed these slides
on counting. In particular, we explained why
(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 $\binom{n}{k}=n!/(k!(n-k)!)$.
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.)
Quiz 8 is at 3-10pm on Canvas!
|
Nov 10
|
We discussed the answers to Quiz 8,
then began a discussion of binomial
coefficients and multinomial coefficients following
these slides.
Read Section 6.2
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Nov 12
|
We reviewed binomial and multinomial coefficients
and their applications. We completed
these slides.
We began to discuss
distribution problems.
Read Section 6.5.
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Nov 15
|
Read Section 6.6.
We reviewed counting formulas, and worked
through most of the exercises on
this handout.
Quiz 9 is at 3-10pm on Canvas!
|
Nov 17
|
Read Section 6.3
We discussed Principle of Inclusion and Exclusion following
this handout.
We solved
one of the problems in class.
We proved the Principle of Incl/Excl by induction
on the number of sets.
|
Nov 19
|
(1) We gave a second proof of the principle of inclusion and exclusion.
(2) The proof in (1) relied on the statement
that the alternating sum of the binomial coefficients
on a single row of Pascal's triangle is zero.
We proved this fact, and noted that it implies that
half of subsets of a $k$-element set have even cardinality and
half have odd cardinality.
(3) We used inclusion/exclusion to show that the number
of surjective functions from an $n$-element set to a $k$-element set
is
$$
\sum_{i=1}^k (-1)^i\binom{k}{i}(k-i)^n
$$
(4) We defined the Stirling numbers of the Second Kind and
proved that $S(n,k)$ is the number of partitions of $n$ into $k$-cells.
(5) We noted that a surjective function is determined by its coimage
and its induced map, and from this it follows that
$$
S(n,k)=\frac{1}{k!}\sum_{i=1}^k (-1)^i\binom{k}{i}(k-i)^n
$$
(6) We started discussing the recursive definition of $S(n,k)$.
|
Nov 29
|
We discussed the recursive definition of $S(n,k)$
and the Bell numbers following
these slides.
Quiz 10 is at 3-10pm on Canvas!
Last One!
Final Exam Review Sheet!
|
Dec 1
|
Discrete probability!
Read Section 6.10.
(Test yourself on the terminology for discrete probability with this
Quizlet link:
https://quizlet.com/646973631/terminology-for-discrete-probability-flash-cards/?x=1qqt.)
Final Exam Review Sheet!
|
Dec 3
|
We reviewed terminology about discrete probability,
and calculated the probabilities for each type of poker hand.
Final Exam Review Sheet!
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Dec 6
|
We reviewed terminology about discrete probability,
and calculated the probabilities for a sequence
of flips of an unfair coin. We also discussed the
problem that arises if one tries to assign
a probability to every subset of an infinite sample space
(Banach-Tarski Paradox).
Final Exam Review Sheet!
|
Dec 8
|
We discussed the Final Exam Review Sheet.
(The final exam will be Saturday, December 11,
4:30-7pm in ECCR 108.)
Final Exam Review Sheet!
|