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.)

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 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 $2+2=4$.

1. We reviewed the difference between sets and classes in terms of the directed graph model of set theory.
2. We explained why we may form empty unions, but not empty intersections. We explained why we may form the intersection of a class of sets but not the union of a class of sets.
3. We defined the natural numbers, $\mathbb{N}$, as the intersection of all inductive sets. We showed that $\mathbb{N}$ is inductive itself, so it is the ``least inductive set''.

Quiz 1!

• class function (used in the Axiom of Replacement)
• choice set (used in the Axiom of Choice)
• enumeration of a set, Well-Ordering Principle (consequence of the Axiom of Choice)
• epsilon-minimal element ($\in$-minimal element) (used in the Axiom of Foundation)

We then reviewed ordered pairs, Cartesian products, and relations.

We turned to new material concerning Functions. The new concepts are described on pages 10-12 of these slides.

1. Function Rule
2. domain, codomain
3. image, preimage, fiber, coimage
4. inclusion map, natural map, induced map

Quiz 2!

1. the function rule
2. the notation $F\colon A\to B$ or $A\stackrel{F}{\to} B$.
3. domain, codomain
4. image, preimage, fiber, coimage
5. inclusion map, natural map, induced map
6. canonical decomposition of a function: $F = \iota\circ\overline{F}\circ \nu$.
7. injection (=injective function), surjection (=surjective function), bijection (=bijective function = injective +surjective function).
8. In response to a comment about coimages, we introduced the concept of a partition of a set $A$. A partition of $A$ is a set $P=\{A_0, A_1, \ldots\}$ of nonempty subsets of $A$ satisfying (i) $\bigcup P=A$ and (ii) $A_i\cap A_j=\emptyset$ when $i\neq j$. Coimages are typical examples of partitions.

1. function (= map)
2. domain, codomain
3. image, preimage, fiber, coimage
4. inclusion map, natural map, induced map
5. canonical decomposition of a function: $F = \iota\circ\overline{F}\circ \nu$.
6. injection (=injective function), surjection (=surjective function), bijection (=bijective function = injective +surjective function).

Next, we turned to the definition of definition. I discussed the three elements of a definition, which answer the questions:

1. What is the word or phrase you are trying to define? (Example: mammal)
2. What type of object is it? To which class does it belong? (Example: member of the animal kingdom)
3. Which properties distinguish your object from others in it class? (Example: a vertebrate animal, warm blooded, gives birth to live young, has body hair, etc)

We explained why the coimage is a partition of the domain and why, conversely, every partition of the domain is a coimage. (So coimages = partitions. More precisely, the concept of a partition is the abstraction of the concept of a coimage.)

I mentioned that it is possible to encode the data of a partition into a binary relation, called an equivalence relation. The equivalence relation associated to $P=\{A_0,A_1,A_2,\ldots\}$ is $K=(A_0\times A_0)\cup(A_1\times A_1)\cup\cdots$. If $P$ is the coimage of $F$, then $K$ is the kernel of $F$.

Quiz yourself on terminology for functions!

Quiz yourself on terminology for binary relations!

Next we discussed kernel of a function/equivalence relation. We defined both and argued that:
the kernel of a function is an equivalence relation and any equivalence relation is a kernel. 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.

Quiz 3!

1. Induction
2. Recursion
3. Arithmetic on $\mathbb N$.

1. Induction review.
2. Addition on $\mathbb N$.

Quiz 4!

1. If the statements to be proved are $S_0, S_1, S_2, \ldots$, then we do not have to prove $S_0$ as the base case. For example, if we prove $S_5$ as the base case, and then we prove the inductive step ($S_k\to S_{k+1}$) for $k\geq 5$, then we will have established that $S_5, S_6, S_7, \ldots$ are true.
2. We introduced and discussed Strong Induction.
3. We examined faulty induction proofs to practice detecting mistakes.

Today we discussed the definitions of finite and infinite, countable and uncountable, and the Pigeonhole Principle following these slides. 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.

Midterm review sheet! (The midterm will be held in class on February 28. You are invited to read this handout on how to answer a question.)

Midterm review sheet! (The midterm will be held in class on February 28. You are invited to read this handout on how to answer a question.)

Quiz 5!

Midterm review sheet! (The midterm will be held in class on February 28. You are invited to read this handout on how to answer a question.)

We began a discussion of logic following these slides. We showed how to write Goldbach's Conjecture as a formal statement. We discussed the first four pages of these slides about Propositional Logic. The topics considered include:

We continued our discussion of propositional logic following pages 5-8 of these slides. The most important topics were:

We began by working on these practice problems. After that, we discussed Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF) following pages 9-13 of these slides Altogether, the practice problems and the DNF/CNF discussion together show that both $A=\{\wedge, \vee, \neg\}$ and $B=\{\to \neg\}$ are `complete' sets of logical connectives. `Completeness' means that any proposition is logically equivalent to an expression using only the connectives in set $A$, and also any proposition is logically equivalent to an expression using only the connectives in set $B$.

• $f\colon \mathbb R\to \mathbb R$ is continuous.
• There are at least three elements.
• There are exactly three elements.
• (About an ordered set) Any two elements have a least upper bound.
• If $a$ is a child of $A$ and $b$ is a child of $B$ and $A$ and $B$ are siblings, then $a$ is a cousin of $b$.

Quiz 6!

Arnie has no respect for those who have no respect for logic.

• To each relation $R\subseteq A^n$ on $A$, there is an associated predicate $P_R\colon A^n\to \{0,1\}$.
• $P_R$ is the characteristic function of $R$.
• $R$ is the support of $P_R$.