Lecture Topics for Discrete Math Students

Course

Math 2001-002: Intro to Discrete Math, Spring 2025

Lecture Topics

Date	What we discussed/How we spent our time
Jan 13	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.)
Jan 15	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 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 $2+2=4$.
Jan 17	We discussed how to enlarge our language through definitions and we introduced definitions for $\emptyset, 0, \subseteq$, and $S(X)$. We introduced the notations $A\cup B$ and $\bigcup X$ for the formation of unions of two or arbitrarily many sets. Note: we may henceforth use the symbols $\cup$ and $\bigcup$. They are defined by $$ \varphi_{((A\cup B)=C)}(A,B,C):\;(\forall z)((z\in C)\leftrightarrow ((z\in A)\vee(z\in B))) $$ and $$ \varphi_{((\bigcup X)=Y)}(X,Y):\;(\forall z)((z\in Y)\leftrightarrow (\exists w)((z\in w)\wedge (w\in X))). $$ We introduced inductive sets. We discussed Axioms 1, 2, 3, 4, and 5 in English, (ii) in terms of diagrams and examples, (iii) in terms of the Directed Graph Model of the Universe of Sets.
Jan 20	MLK, Jr Day! No meeting.
Jan 22	We reviewed the material from last week and introduced the Axiom of Power Set and the Axiom of Separation. We discussed the relationship between Restricted Comprehension and Unrestricted Comprehension. Our discussion covered pages 1-8 of these slides, pages 1-4 of these slides, and pages 1-3 of these slides.
Jan 24	We discussed why Naive Set Theory is inconsistent following these slides. We introduced the concept of a class. We explained why every set is a class, but not every class is a set. (The Russell class is an example of a class that is not a set.) If $x$ and $y$ are sets and $X$ and $Y$ are classes, then it is meaningful to write $x\in y$ and $x\in Y$, but not $X\in y$ and not $X\in Y$. That is, a proper class should not appear to the left of the symbol $\in$. We explained why there is no set of all sets. 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. Here intersection behaves differently than union: $\bigcup\emptyset$ IS a set.
Jan 27	We reviewed the difference between sets and classes in terms of the directed graph model of set theory. 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. 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 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!
Jan 29	We completed the introduction of the ZFC axioms by discussing the last three axioms (Replacement, Choice, Foundation). Some of the new terminology introduced is: 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)
Jan 31	We discussed Ordered Pairs.