Date
|
What we discussed/How we spent our time
|
Aug 28
|
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 to
https://quizlet.com/join/mExWGGZqj.
(Test yourself on the Axioms of Set Theory with this
Quizlet link:
https://quizlet.com/_61ko6h.)
|
Aug 30
|
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 language of set theory.
We described the directed graph model of set theory. We
introduced
(i) the symbol $\in$.
(ii) the Axiom of Extensionality.
(iii) the Axiom of the Empty Set.
(iv) the Axiom of Pairing.
|
Sep 1
|
We discussed how to enlarge our language through definitions,
and we introduced definitions for $\emptyset, \subseteq$,
and ${\mathcal P}(X)$.
We discussed Axioms 1, 3, 4, 5, and 6 (i) in English, (ii)
in terms of Venn diagrams and examples, (iii) in terms
of the Directed Graph Model of the Universe of Sets,
and (iv) in the formal language of Set Theory.
|
Sep 6
|
We took a
practice quiz.
Then we introduced the successor operation $S(x)=x\cup \{x\}$
and used it to define inductive sets.
We stated the Axiom of Infinity.
|
Sep 8
|
We discussed the Axiom of Separation.
We discussed Unrestricted Comprehension versus Restricted Comprehension.
We explained why naive set theory is inconsistent
and explained how naive set
theory differs from axiomatic set theory .
We explained
the difference between sets and classes,
and discussed some examples of proper classes.
(The Russell class, the class of all sets,
the class of all $1$-element sets.)
|
Sep 11
|
We completed the introduction of the ZFC axioms
by discussing the last three axioms
(Replacement, Choice, Foundation).
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 1!
|
Sep 13
|
We discussed the first 6 pages of these
notes on Ordered Pairs.
|
Sep 15
|
We discussed pages 6-9 of these
notes on Ordered Pairs.
|
Sep 18
|
We completed these
notes on Ordered Pairs. The main topics were:
Directed graph representations of binary relations.
Functions:
- Function Rule
- domain, codomain
- image, preimage, fiber, coimage
- inclusion map, natural map, induced map
Quiz yourself
on terminology for functions!
Quiz 2!
|
Sep 20
|
Today we defined
The kernel of a function.
(If $F\colon A\to B$ is a function, then
$\ker(F)=\{(a,a')\in A\times A\;|\;F(a)=F(a')\}$.)
Equivalence relation.
(= a reflexive, symmetric, transitive binary relation.)
Equivalence class. ($[a]=[a]_E=\{b\in A\;|\;(a,b)\in E\}$)
The set of $A/E=\{[a]_E\;|\;a\in A\}$ of $E$-equivalence classes.
We read $A/E$ as ``$A$ modulo $E$'' or ``$A$ mod $E$''.
The natural map associated to an equivalence relation. ($\nu(a)=[a]_E$)
We proved that a binary relation $E\subseteq A\times A$
is a kernel of a function with domain $A$
if and only if it is
an equivalence relation on $A$.
The proof showed that any
equivalence relation $E$ on a set $A$ is the kernel of
the natural map $\nu\colon A\to A/E\colon a\mapsto [a]_E$
that is associated to $E$.
|
Sep 22
|
Today we discussed this handout
about function terminology. We spent some time relating
the four concepts:
kernel of a function/equivalence relation
coimage of a function/partition
The main elements of the relationship are these:
The kernel of a function is an equivalence relation.
Any equivalence relation is the kernel of a function.
Hence the definition of ``equivalence relation''
correctly axiomatizes kernels.
The coimage of a function is a partition.
Any partition is the coimage of a function.
Hence the definition of ``partition''
correctly axiomatizes coimages.
Equivalence relations on $A$ and partitions of $A$
``carry the same information''.
The main difference is that equivalence relations
are sets of pairs, while partitions are sets of sets.
|
Sep 25
|
We reviewed function terminology,
and discussed when a function
is injective or surjective.
Quiz 3!
|
Sep 27
|
Today we completed this handout
about function terminology.
We spent most of the time
discussing injective, surjective, bijective,
and invertible functions along
with composition of functions.
We discussed the injectivity/surjectivity of these examples (among others):
$f\colon \mathbb R\to \mathbb R\colon x\mapsto e^x$,
$g\colon \mathbb R\to \mathbb R\colon x\mapsto x\cdot \sin(x)$,
$h\colon \mathbb R\to \mathbb R\colon x\mapsto x+\sin(x)$.
We explained why a function is injective
if and only if it is left cancellable if and only
if its kernel is as small as possible (equal to the equality relation).
We explained why a function is surjective if
and only if it is right cancellable if and only if its image
is as large as possible (equal to the codomain).
|
Sep 29
|
We
defined the natural numbers
as the intersection of all inductive sets.
We proved that the set of natural numbers is inductive itself,
hence it is the ``least inductive set''.
We discussed the definitions of the order
relation and the arithmetic on $\mathbb N$.
We discussed the Principle of Induction
following
these slides.
Review sheet for the midterm.
|
Oct 2
|
We discussed (the Parametrized version of) the Recursion Theorem.
We applied it to define $x+y, x\cdot y$ and $x^y$
on $\mathbb N$. We discussed how to prove
the Laws of Arithmetic in
this handout.
Review sheet for the midterm.
Quiz 4!
|
Oct 4
|
We reviewed for the midterm following
this review sheet.
(The midterm will be held in class on October 6.)
We discussed this handout
on how to answer a question.
We also solved Problem 1 of HW4 as an example
of a proof by induction.
|
Oct 6
|
Midterm!
Midterm Answer Key
|
Oct 9
|
Read Section 2.7.
We began to discuss cardinality
following this handout.
|
Oct 11
|
Read Section 2.7.
Today we discussed
the definitions of
finite and infinite, the Pigeonhole Principle,
and we explained why $\mathbb N$ is infinite.
While discussing these topics, we introduced
the concept of the restriction
of a function $f\colon A\to B$ to a subset
$A'\subseteq A$ of the domain.
|
Oct 13
|
We discussed the disjoint union
construction: given sets
$A$ and $B$, we construct
$$\widehat{A}=\{(a,0)\in A\times\{0\}\;|\;a\in A\}$$
and
$$\widehat{B}=\{(b,1)\in B\times\{1\}\;|\;b\in B\}.$$
There exist bijections
$f\colon A\to \widehat{A}\colon a\mapsto (a,0)$
(so $|A|=|\widehat{A}|$)
and
$g\colon B\to \widehat{B}\colon b\mapsto (b,1)$
(so $|B|=|\widehat{B}|$)
and we have $\widehat{A}\cap \widehat{B}=\emptyset$.
We showed that $|A|=|A'|, |B|=|B'|$ and $|A|\leq |B|$
imply $|A'|\leq |B'|$.
We discussed and prove the Cantor-Bernstein-Schröder Theorem
and used it to show that $|\mathbb N\times \mathbb N|=|\mathbb N|$.
|
Oct 16
|
We proved Cantor's Theorem.
Quiz 5!
|
Oct 18
|
We explain why $|\mathbb R| = |(0_{\mathbb R},1_{\mathbb R})| =
|{\mathcal P}(\mathbb N)|$. Then we sketched
how the integers are constructed from the natural numbers.
|
Oct 20
|
Read Section 3.1.
We began a discussion of logic following
some handouts logic.pdf,
logic2.pdf, and
prac1.pdf.
We showed how to write Goldbach's Conjecture as a formal statement.
We discussed the alphabet of symbols for a formal statement and
how to draw
the parse tree (=formula tree) for a formal statement.
We discussed the first four
pages of these slides
about Propositional Logic.
These pages discuss truth tables for the propositional
connectives $\wedge, \vee, \neg, \to ,\leftrightarrow$,
and how to determine the truth table
for a compound proposition.
We defined tautology and contradiction.
Quiz yourself
on terminology for Propositional Logic!
XKCD on Mathematical Symbols.
|
Oct 23
|
We reviewed these slides
about Propositional Logic up to Slide 8.
The topics considered include:
Propositions. Propositional variables. Compound propositions.
Logical connectives.
Truth tables.
Tautologies. Contradictions. Logical equivalence.
Disjunctive Normal Form.
We explained how to convert any proposition
into an equivalent proposition written in
Disjunctive Normal Form.
Quiz yourself
on terminology for Propositional Logic!
Quiz 6!
|
Oct 25
|
We completed these slides
about Propositional Logic. New topics included:
Logical equivalence, logical inequivalence, logical independence.
A comparison of: direct implication, contrapositive implication,
converse implication, inverse implication.
Disjunctive Normal Form. Completeness of
any set of connectives containing AND, OR, and NOT.
DNF depends on the set of variables.
We completed these slides
about predicates (one definition + examples).
We began this handout
on quantifiers.
Arnie has no respect for those who have no respect for logic.
|
Oct 30
|
We gave the recursive definition
of formula.
We practiced determining tables for atomic formulas
following this handout.
Solutions!
Quiz 7!
|
Nov 1
|
The due date for HW7 was moved from Nov 1 to Nov 8.
We completed the discussion of how to determine
whether a sentence is true in a structure.
Most of the time was spent following
this handout,
which discusses how to handle quantifiers.
We completed the first two pages of this handout
on quantifiers.
This handout was circulated, but
we didn't get to work on it.
|
Nov 3
|
The due date for HW7 was moved from Nov 1 to Nov 8.
We discussed prenex form following
these slides.
We then discussed part of page 3 of
this handout.
|
Nov 6
|
We discussed quantifier games
and worked through some problems
on this handout.
Quiz 8!
|
Nov 8
|
We discussed truth versus provability.
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) Silly proof systems. (The empty proof system, the proof system where every sentence in the language is an axiom.)
(6) Soundness and completeness of a proof system.
(7) First-order sentences.
(8) Gödel's Completeness Theorem.
(9) The structure of a theorem statement.
(10) Proof strategies.
(11) Direct proof, proof of the contrapositive,
and proof by contradiction.
|
Nov 10
|
We reviewed the material from the previous lecture.
We then extended the material in the following ways:
(1) We discussed `mixed' proof strategies.
For example, we indicated that a theorem of the form
$(H_1\wedge H_2)\to C$ could be proved in any of the following ways:
$H_1, H_2, S_3, \ldots, S_k, C$
$H_1, \neg C, S_3, \ldots, S_k, \neg H_2$
$H_1, H_2, \neg C, S_4, \ldots, S_k, \bot$
(2) We discussed how to handle quantifiers in proofs.
The example proofs we considered were:
Theorem.
For any nonempty structure,
$(\forall x)\varphi(x)$ implies $(\exists x)\varphi(x)$.
Theorem.
Suppose that $f\colon A\to B$ and $g\colon B\to C$ are functions.
If $(\forall b)(\exists a)(f(a)=b)$ and
$(\forall c)(\exists b)(g(b)=c)$, then
$(\forall c)(\exists a)(g\circ f(a)=c)$.
(3) We discussed `restricted quantifiers', like the
ones that appear in the definition $\lim_{x\to a} f(x) = L$. This definition is
$(\forall \varepsilon >0)(\exists \delta > 0)(0<|x-a|<\delta\to |f(x)-L|<\varepsilon).$
The expression $(\forall \varepsilon >0)$ means that we are universally quantifying $\varepsilon$ subject to the restriction $\varepsilon > 0$, and similarly the expression $(\exists \delta >0)$ means that we are existentially quantifying $\delta$ subject to the restriction $\delta > 0$.
The two main things to remember from this are that (i) sentences with restricted
quantifiers, like $(\forall x P(x))Q(x)$ or $(\exists x P(x))Q(x)$, are abbreviations for sentences that involve only
unrestricted quantifiers:
$(\forall x P(x))Q(x) \equiv (\forall x)(P(x)\to Q(x))$ and
$(\exists x P(x))Q(x)\equiv (\exists x)(P(x)\wedge Q(x))$,
and (ii) restricted quantifiers follow the same rules as
unrestricted quantifiers. For example,
$\neg (\forall x P(x))Q(x)\equiv (\exists x P(x))\neg Q(x)$,
$\neg (\exists x P(x)) Q(x)\equiv (\forall x P(x)) \neg Q(x)$, and (for example) $R\wedge ((\forall x P(x)) Q(x)\equiv (\forall x P(x) (R\wedge Q(x))$ if $R$ does not depend on $x$.
|
Nov 13
|
Read Section 6.1
We began discussing counting following
these slides.
Today we only covered the first three slides.
Quiz 9!
|
Nov 15
|
We continued discussing
these slides.
Today we got to the first item on the last slide.
Review sheet for the final exam.
|
Nov 17
|
Read Section 6.2.
We finished
these slides.
We began
discussing the Binomial Coefficients.
We discussed:
The definition:${n \choose k}=$ the number of $k$-element subsets
of an $n$-element set.
A formula for ${n \choose k}$.
A recursion for computing ${n \choose k}$.
Pascal's Triangle.
$\sum_{k=0}^n {n \choose k} = 2^n$.
Review sheet for the final exam.
|
Nov 27
|
Read Section 6.6.
We reviewed binomial
coefficients and the binomial theorem
and then
introduced
multinomial coefficients and the multinomial
theorem.
We introduced the concept of a multiset.
We followed
these slides
up to the middle of slide 8.
Quiz 10!
|
Nov 29
|
We reviewed the solutions to Quiz 10
(with special attention to `and' versus `or').
We discussed all formulas on the
distributions handout
except the formula $n!\cdot S(k,n)$.
Solutions to the
problems on the handout!
Quiz yourself
on Counting Formulas!
Review sheet for the final exam.
|
Dec 1
|
We reviewed the distribution formulas.
We reviewed the answers to the problems
on the distributions handout.
We started discussing the principle of inclusion and exclusion.
Review sheet for the final exam.
|
Dec 4
|
We
generalized the Principle of Inclusion
and Exclusion so that it applies to
more than $2$ sets. The proof of the generalization
depended on a fact about binomial coefficients:
$$
{k \choose 0} -
{k \choose 1} +
{k \choose 2} - \cdots + (-1)^k{k \choose k} = 0.
$$
We used the Inclusion/Exclusion formula to count the number
of surjective functions $f\colon 5\to 3$.
Quiz 11!
|
Dec 6
|
We discussed some problems from the
Inclusion/Exclusion handout.
We showed that the number of surjective functions
from an $n$-element set to a $k$-element set is
$$
\sum_{r=0}^k (-1)^r{k \choose r}(k-r)^n.
$$
From this we derived that the number of partitions
of an $n$-element into $k$ cells is
$$
\frac{1}{k!}\sum_{r=0}^k (-1)^r{k \choose r}(k-r)^n.
$$
We started discussing the Stirling numbers of the second kind
following these slides.
You can learn about Stirling numbers of the first kind
here, but we will not discuss them.
Review sheet for the final exam.
|
Dec 8
|
We completed the slides on the
Stirling numbers
of the second kind and the Bell numbers. The material is
summarized on this handout, but
the handout has some extra practice problems.
The most important definitions are:
$S(n,k)$ denotes the number of partitions of
an $n$-element set into $k$ cells. (This is a Stirling
number of the second kind.)
$B_n$ denotes the number of partitions of an $n$-element set.
($B_n$ is a Bell number.)
Review sheet for the final exam.
|
Dec 11
|
We worked on
practice problems.
Review sheet for the final exam.
|
Dec 13
|
We discussed the
Review sheet for the final exam.
Some solution sketches for the review sheet.
The Final Exam will be held at December 17, 1:30-4pm (Sunday),
in our usual classroom DUAN G2B41.
|