The main goals of the course are defined 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 ``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.)

We discussed the Axiom of Choice and the Axiom of Regularity/Foundation. In the discussion of the Axiom of Foundation we defined the notion of an $\in$-minimal element of a set.
We defined ZFC (all 10 axioms) and ZF (all 10 axioms minus the Axiom of Choice).
We discussed classes (like the class of all sets), explaining what classes are and how they may differ from sets. We showed that the Axiom of Separation allows us to intersect any nonempty class of sets.

Despite this duality, we noted that there are some asymmetries between union and intersection. The first asymmetry we noted was that $\bigcup \emptyset$ is a valid set (it is $\emptyset$), but $\bigcap \emptyset$ is not set. The second asymmetry we noted about union and intersection is that we can only form the union of a collection that is a set, but we can form the intersection of any collection that is a nonempty class. It is important that we can form these types of intersections, since the set of natural numbers, $\mathbb N$, is defined as the intersection of the class of inductive sets. (We explained today why the class of inductive sets is not a set.)

Finally, we introduced the Kuratowski encoding of ordered pairs, namely $(a,b) := \{\{a\},\{a,b\}\}$. We stated that, with this definition the following theorem holds:

Theorem. $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$. (Not proved yet!)

(Test 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.)

Theorem. $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$.

We defined ordered triple, ordered $n$-tuple, and Cartesian product $A\times B$. We explained why, if $A$ and $B$ are sets, then $A\times B$ is also a set. Quiz 1.

We defined relations and explained the connection between relations and predicates. Handout on relations.

We defined functions and went over some vocabulary for functions. Quiz yourself!

What type of mathematical object is a domain, codomain, image, or coimage?

Any domain is a set and any set is a domain. Any codomain is a set and any set is a codomain. We defined the identity function on $S$, ${\rm id}_S:S\to S$, to show how to realize any set $S$ as a domain or a codomain.

The image of a function is a subset of the codomain. Conversely, if $B$ is any set and $S\subseteq B$ is any subset, then there is a function, $\iota_S:S\to B$, the inclusion function for $S$ into $B$, for which $B$ is the codomain and $S$ is the image.

The coimage of a function is a partition of the domain. Conversely, if $A$ is any set and $P$ is any partition of $A$, then there is a function, $N_P:A\to P: a\mapsto [a]$, the natural map for $P$, for which $A$ is the domain and $P$ is the coimage.

Quiz 2.

Directed graph representation of binary relations. Equivalence relations are the abstraction of kernels. Quiz yourself!

Recursion and induction, I. Exploiting the defining property of $\mathbb N$. Why is recursion a valid way to define a function? Why is induction a valid form of proof?

SMBC on induction.

First logic handout. Structures. Alphabet of symbols. Ingredients in a compound predicate.

No quiz!

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

Second logic handout. We gave a recursive definition of the set of terms (or algebraic expressions) in some language. Then we discussed how to give meaning (= assign tables) to terms.

Third logic handout. Tables for the logical connectives. Tautology and contradiction. Logical equivalence of propositions. Quiz yourself on propositional logic!

We proved the additive and multiplicative counting principles, using induction.

We defined the numbers $C(n,k) = {n \choose k}$ combinatorially. We showed that these numbers may be defined recursively. We showed that these are the numbers that arise in Pascal's Triangle. We showed that (i) the sum of the numbers in the $n$-th row of Pascal's triangle is $2^n$, (ii) the alternating sum of the numbers in the $n$-th row of Pascal's triangle is $0$ if $n>0$, (iii) Pascal's Triangle is symmetric about its main diagonal, and (iv) we indicated how to prove that the $n$-th row is a unimodal sequence.

We gave two proofs of the binomial theorem. Then we defined trinomial coefficients, gave a formula for them, explained the recursion for them (Pascal's Pyramid), and stated the trinomial theorem. We indicated how to generalize to multinomial coefficients. Quiz 9.

Selections with repetitions. We introduced multisets and multichoose coefficients: $MC(n,k) = \left({n \choose k}\right)={{n+k-1}\choose k}$.

Inclusion-exclusion. Counting surjective functions.

Quiz yourself on counting formulas!