Date
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What we discussed/How we spent our time
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Aug 21
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Syllabus. Policies. Text.
The use of AI in this class.
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Aug 24
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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.)
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Aug 26
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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 discussed
(i) the symbol $\in$.
(ii) the Axiom of Extensionality.
(iii) the Axiom of the Empty Set.
(iv) the Axiom of Pairing.
(iv) the Axiom of Union and the definition of 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 28
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We discussed expanding the language of set
theory following
these slides.
This included a discussion of
the subset relation and the power set function.
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Aug 31
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We discussed
(i) the directed graph model of set theory.
(ii) the Axiom of Separation.
(iii) the fact that Naive Set Theory is inconsistent
(Russell's Paradox).
Quiz 1 solutions.
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Sep 2
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We reviewed Unrestricted Comprehension versus Restricted Comprehension,
Russell's paradox, and the fact that naive set theory is inconsistent.
Then we discussed
(i) why there is no set of all sets.
(ii) why we do not need an Axiom of Intersection.
(iii) the
Axioms of Replacement, Choice, and Foundation.
(iv) how the Axiom of Replacement can be used in place
of the Axiom of Separation to separate out nonempty subsets
of a given set.
(v) the fact that the Axiom of Choice is equivalent over ZF to the
statement that every set can be enumerated y an ordinal.
(vi) the fact that the Axiom of Foundation implies that the
successor function is injective.
We concluded by defining the symbols
$\cap$ and $\bigcap$ for intersection
and we stated without proof that
the class $\bigcap \emptyset$ is not a set.
In this way, intersection behaves differently than union:
$\bigcup\emptyset$ IS a set. (Check: $\bigcup\emptyset=\emptyset$.)
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Sep 4
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We discussed Ordered Pairs and Relations.
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Sep 7
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Labor Day! No class.
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Sep 9
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We discussed
Functions and Operations.
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Sep 11
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We continued
these slides.
Some details are expanded in
this handout.
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Sep 14
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We finished
these slides.
Some details are expanded in
this handout.
The main new concept was the kernel of a function.
Quiz 2 solutions.
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Sep 16
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We spent some time on a group quiz.
Then we discussed how to answer a question.
We concluded the meeting by reviewing the definition
of the kernel of a function.
We explained how to determine the kernel of a function
if we are given the coimage of the function.
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Sep 18
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The abstraction of the concept of a kernel is
the concept of an equivalence relation.
We defined equivalence relations and showed
that every kernel is an equivalence relation
and every equivalence relation is a kernel.
We also explained how to determine
the coimage associated to a given kernel.
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Sep 21
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We discussed inductive sets and
the definition of the natural numbers
following these slides.
We showed that the natural numbers is the least inductive set.
Quiz 3 solutions.
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Sep 23
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We reviewed Quiz 3, emphasizing how to express some
concepts. For example, if $E\subseteq A\times A$,
then how do we correctly express the properties
(i) $E$ is reflexive, (ii) $E$ is symmetric, (iii)
$E$ is transitive? We mentioned that, if $E$
is an arbitrary binary relation
on $A$ and $(a,b)\in E$,
then one should say that
``$a$ is related to $b$'',
``$a$ is $E$-related to $b$'', or
``$E$ relates $a$ to $b$'',
but not
``$E$ maps $a$ to $b$''.
A map is a function, so if $E$ is not a function,
then it cannot map elements from one place to another.
We next began discussing induction, recursion, and the
arithmetic
of $\mathbb{N}$ following
these slides.
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Sep 25
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The main topics today were
- Induction
- Recursion
- Arithmetic on $\mathbb N$.
We completed the first 10 pages of
these slides.
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Sep 28
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We completed the proof of the Commutative Law
for addition of natural numbers.
Then we discussed
Strong Induction
and we used it tp prove that every
positive natural number is a product of primes.
Quiz 4 solutions.
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Sep 30
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We recalled the definition of Strong Induction
and stated the definition of
Course-of-Values Recursion.
We used C-o-V Recursion to define the Fibonacci numbers.
Next, we discussed examples of
mistakes in induction proofs.
Midterm 1 Review Sheet.
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