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Math 2001-003: Intro to Discrete Math, Fall 2021


Lecture Topics


Date
What we discussed/How we spent our time
Aug 23
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 https://quizlet.com/join/mExWGGZqj.

(Test yourself on the Axioms of Set Theory with this Quizlet link: https://quizlet.com/_61ko6h.)

Aug 25
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 directed graph model of ZFC. We introduced
(i) the symbol $\in$.
(ii) the Axiom of the Empty Set.
(iii) the Axiom of Extensionality.
(iv) 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 $2+2=4$.

Aug 27
We reviewed naive set theory versus formal set theory. We discussed four of the first five axioms of set theory (Extensionality, Empty Set, Pairing, Union). We discussed the use of definitions/abbreviations to introduce new symbols into mathematics (like the symbols $\emptyset, \bigcup, S(x)$).
Aug 30
We defined inductive set. We introduced the Axiom of Infinity and the Axiom of Restricted Comprehension. We defined $\mathbb N$ to be the set of elements common to all inductive sets. We proved that $\mathbb N$ is an inductive set, and noted that it is the intersection of all inductive sets, and therefore is the least inductive set.
Sep 1
We introduced the concept of a subset of a set, defined the notation $x\subseteq y$, and stated the Power Set Axiom. We proved that $A=B$ if and only if $A\subseteq B$ and $B\subseteq A$. We then proved the distributive law $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$.
Sep 3
We discussed the following questions.
(i) What is naive set theory?
(ii) Why is naive set theory inconsistent?
(iii) Why should we have expected naive set theory to be inconsistent?
(iv) Do the axioms of ZFC protect us from contradictions?
(v) What kind of object is $\{x\;|\;x\notin x\}$? Is it still possible to discuss such objects mathematically?

Our discussion involved defining the Russell class, a discussion of Russell's Paradox, Gödel's Second Incompleteness Theorem, and classes versus sets.

We spent the last 15 minutes of class discussing the statements and meanings of the Axioms of Replacement, Choice, and Foundation.

Sep 8
We introduced the Kuratowski encoding of ordered pairs: $(a,b)=\{\{a\},\{a,b\}\}$.
(i) We explained why $(a,b)$ is a set.
(ii) We proved that $(a,b)=(c,d)$ implies $a=c$ and $b=d$.
(iii) We defined ordered triples and quadruples
(iv) We defined the Cartesian product $$A\times B = \{x\in{\mathcal P}{\mathcal P}(A\cup B)\;|\;x=(a,b), a\in A, b\in B\}$$ of sets $A$ and $B$. We use the product symbol $\times$ for Cartesian products when using infix notation, $A_1\times A_2\times A_3\times A_4$, and the symbol $\prod$ when using prefix notation, $\prod_{i=1}^4 A_i$. If all the factors are the same, we might write $A^4$ for $A\times A\times A\times A$.
Sep 10
Read Sections 2.2 and 2.3.

We discussed how ordered pairs and $n$-tuples are used to define relations and functions. We discussed the notation $$f\colon A\to B\colon x\mapsto x^2.$$ We discussed the following terminology about functions:
(1) Directed graph representation of a function.
(2) Domain of a function.
(3) Codomain of a function.
(4) Image of a function.
(5) Preimage of an element under a function.
(6) Fiber over an element.
(7) Coimage of a function.
(8) Inclusion function. (Or inclusion map.)
(9) Natural function. (Or natural map.)
(10) Induced function. (Or induced map.)
(11) Canonical factorization of a function.

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 yourself on terminology for functions!

Sep 13
We reviewed all of the function terminology, and saw how to write each concept down for the squaring function $$F\colon \mathbb R\to \mathbb R\colon x\mapsto x^2.$$ In addition, we defined
(1) Composition of functions.
(2) Injective function.
(3) Surjective function.
(4) Bijective function.
(5) Identity function.
(6) Inverse of a function.

Quiz 1 is at 5-7pm on Canvas!

Sep 15
We explained why the natural map is surjective, but usually not injective. We explained why the inclusion map is injective, but usually not surjective. We explained why the induced map is bijective.

We defined kernel, partition, and equivalence relation, and explained the relationship coimages$\leftrightarrow$partitions, namely the coimage of a function with domain $A$ is a partition of $A$, while any partition of $A$ is the coimage of a function with domain $A$.

Sep 17
Read Section 2.5.

We began to explain the relationship kernels$\leftrightarrow$equivalence relations, namely the kernel of a function with domain $A$ is an equivalence relation on $A$, while any equivalence relation on $A$ is the kernel of a function with domain $A$.
In this explanation, we introduced the concept of equivalence class. Here, if $E$ is an equivalence relation on $A$, the set $[a]_E=\{x\in A\;|\;(a,x)\in E\}$ is the $E$-equivalence class of $a$. The set $A/E=\{[a]_E\;|\;a\in A\}$ is a partition of $A$ into $E$-equivalence classes.

Quiz yourself on coimages, kernels, partitions and equivalence relations with this Quizlet link: https://quizlet.com/371093838/

Sep 20
We reviewed the concept of ``well foundedness'' and how the definitions of a ``function'', ``number'', and ``finite''/``infinite'' are developed from the axioms.

We began discussing induction, recursion, and the arithmetic of $\mathbb N$. (Some hints.)

Quiz 2 is at 5-7pm on Canvas!

Sep 22
We discussed examples of definition by recursion and proof by induction. (This included formulas for the sum of the first $n$ positive integers, the sum of the first $n$ positive odd integers, the sum of the first $n$ squares, the sum of the first $n$ cubes, and the formula for the sum of a geometric series.)
Sep 24
We discussed other forms of recursion (recursion using parameters and course-of-values recursion). Then we proved the laws of successor and the laws of addition from the handout on the arithmetic of $\mathbb N$.
Sep 27
We discussed
(1) Induction starting at $s_n$ instead of $s_0$.
(2) False induction proofs.
(3) Course-of-values induction.

The example proofs we discussed were
(1) A correct proof by ordinary induction that $2^n\leq n!$ for $n\geq 4$.
(2) A false proof by ordinary induction that when you put $n$ points on the boundary of a disk and cut the disk through all pairs you obtain $2^{n-1}$ regions.
(3) A false proof by ordinary induction that all horses have the same color.
(4) A correct proof by strong induction that every positive number is a product of prime numbers.

Quiz 3 is at 5-7pm on Canvas!

Midterm Review Sheet!

Sep 29
Read: Sections 2.7 and 2.8.

We started discussing ordinal and cardinal numbers.

We defined ordinal numbers and introduced the symbol $\omega$ (omega) as a symbol for the first infinite ordinal. $\omega=\mathbb N$. We discussed the main properties of ordinals, including that ordinals are well ordered and that every set can be enumerated by an ordinal. We showed that $\omega$ can be enumerated by each of $\omega, \omega+1, \omega+2$ and $\omega+\omega$. We then defined equipotence, finite, infinite, countably infinite, countable, and uncountable. We introduced the notation $|A|=|B|$, $|A|\leq |B|$, and $|A|<|B|$.

Quiz yourself on ordinals and cardinals with this Quizlet link: https://quizlet.com/375068176/

Midterm Review Sheet!

Oct 1
We continued discussing ordinal and cardinal numbers.

We proved the Cantor-Bernstein-Schroeder Theorem. We derived the Corollary that $A\subseteq B\subseteq C$ and $|A|=|C|$ together imply that $|A|=|B|=|C|$. Hence equipotence classes of ordinals are convex. We showed that $|\mathbb N|=|\mathbb N\times \mathbb N|$ in two ways, one way using the CBS theorem and one way not using it.

Midterm Review Sheet!

Oct 4
We continued discussing ordinal and cardinal numbers.

We proved

(1) If $A$ is an alphabet satisfying $|A|\leq |\mathbb N|$ and each element of a set $X$ can be described by a finite-length sentence in the alphabet $A$, then $|X|\leq |\mathbb N|$.
(2) $|\mathbb Z|=|\mathbb N|$. (Describe integers with alphabet $A=\{-\}\cup \mathbb N$.)
(3) $|\mathbb Q|=|\mathbb N|$. (Describe rational numbers with alphabet $A=\{-, /\}\cup \mathbb N$.)
(4) $|\mathbb R|\leq |(0,1)|\leq |{\mathcal P}(\mathbb N)|\leq |\mathbb R|$.
(5) Cantor's Theorem.
(6) $|\mathbb N|\lt |{\mathcal P}(\mathbb N)|=|\mathbb R|$.

We also mentioned that $\omega_0$ is notation for the first countable ordinal/cardinal, and $\omega_1$ is notation for the first uncountable ordinal/cardinal.

Quiz 4 is at 5-7pm on Canvas!

Midterm Review Sheet!

Oct 6
We reviewed for the midterm. (The midterm will be in-class on Friday October 8.)

Midterm Review Sheet! Some hints!

Oct 8
Midterm! Solutions!
Oct 11
Read: pages 89-95.

We are starting to study logic (Chapters 3 and 4). In Chapter 3, we will discuss material from Sections 3.1 and 3.5, and part of 3.2. We will not discuss material from 3.3, 3.4, or 3.6.

Today we discussed two introductory handouts on logic. We introduced the notions of proposition (= ``declarative sentence'', or ``sentence which can be assigned a truth value''), propositional connective, and the truth tables for $\wedge, \vee, \neg, \to, \leftrightarrow$.

No quiz!

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

Oct 13
No HW due today! (New assignment posted for next week!)

Today we discussed propositional logic following these slides. (We got to page 6.)

Oct 15
Today we completed these slides. Truth table review and practice! Hints!
Oct 18
Read Subsection 4.1.3.

We discussed more propositional logic, including
(1) the equivalences $H\to C\equiv (\neg C)\to (\neg H)\equiv (H\wedge (\neg C))\to \textrm{False}$.
(2) the contrapositive, converse, and inverse of $H\to C$.
(3) logical consequence and logical independence. (We say that $Y$ is a logical consequence of $X$ if $X\to Y$ is a tautology. $X$ and $Y$ are logically independent if neither is a logical consequence of the other.)

We talked about structures and their structural elements.

Quiz 5 is at 3-10pm on Canvas!
(This is a quiz on Propositional Logic. See the lecture notes from October 11-15.)

Oct 20
The due date for HW6 was moved to October 27.

Today we defined Terms, Atomic Formulas, Formulas, We drew a Term Tree and a Formula tree.

Example: in the formula $$ (\forall a)(\exists b)(\forall c)(\exists d)((a^2+b^2=c^2+d^2)\vee (a^2+c^2=b^2+d^2)) $$ the atomic formulas are $F=(a^2+b^2=c^2+d^2)$ and $G=(a^2+c^2=b^2+d^2)$. The expressions $a^2+b^2,c^2+d^2,a^2+c^2$, and $b^2+d^2$ are examples of terms. The formula has the logical structure $$ (\forall a)(\exists b)(\forall c)(\exists d)(F\vee G) $$

Oct 22
The due date for HW6 has been moved to October 27.

Today we discussed tables for structural elements. We worked on these practice problems. Solutions! We discussed the first 1.5 pages of this handout on quantifiers.

Oct 25
The due date for HW6 has been moved to October 27.

Today we completed this handout on quantifiers. (We solved Practice Problem 5 in class.)

Quiz 6 is at 3-10pm on Canvas!

Oct 27
We discussed
(1) how the meaning of a sentence in a structure is determined by the tables of the structural elements. (This was review from Oct 18-22!)
(2) how one can use quantifier games to determine truth or falsity. (This was review from Oct 25!)
(3) how to put a sentence in prenex form. (This was new material!)
Oct 29
We finished discussing the prenex form handout. In particular, we discussed how to standardize the variables apart in a sentence and then to put the sentence in prenex form.
Nov 1
We reviewed and practiced what it means for a sentence to be true in a structure. Then we discussed semantic consequence. ($\Sigma \models P$.) We discussed provability versus truth. 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) Soundness and completeness of a proof system.
(6) First-order sentences.
(7) Gödel's Completeness Theorem.

Quiz 7 is at 3-10pm on Canvas!

Nov 3
We discussed
(1) The structure of a theorem statement.
(2) Proof strategies.
(3) The equivalence of direct proof, proof of the contrapositive, and proof by contradiction.
(4) Proofs involving quantifiers.
Nov 5
We discussed Formal versus informal proofs. Then we began a discussion of counting. We stated the Sum Rule and the Product Rule, and proved the Sum Rule.

Read Section 6.1.

Nov 8
We discussed these slides on counting. In particular, we explained why

(1) an $n$-element set has $2^n$ subsets,
(2) the number of functions from a $k$-element set to an $n$-element set is $n^k$,
(3) the number of bijective functions from a $k$-element set to an $n$-element set is $0$ is $k\neq n$ and is $n!$ if $k=n$,
(4) the number of injective functions from a $k$-element set to an $n$-element set is $(n)_k=n!/k!$, and
(5) the number of $k$-element subsets of an $n$-element set is $\binom{n}{k}=n!/(k!(n-k)!)$.

During this discussion, we discussed overcounting, in the following form: If $E$ is an equivalence relation on $L$, and all $E$-classes have the same size $k$, then the number of $E$-classes is $|L|/k$. (Special case: You can count the number of cows in a field by counting their legs and dividing by four. Here $L$ is the set of legs, and two legs are called equivalent if they belong to the same cow. We assume that each cow has $k=4$ legs.)

Quiz 8 is at 3-10pm on Canvas!

Nov 10
We discussed the answers to Quiz 8, then began a discussion of binomial coefficients and multinomial coefficients following these slides.

Read Section 6.2

Nov 12
We reviewed binomial and multinomial coefficients and their applications. We completed these slides. We began to discuss distribution problems.

Read Section 6.5.

Nov 15
Read Section 6.6.

We reviewed counting formulas, and worked through most of the exercises on this handout.

Quiz 9 is at 3-10pm on Canvas!

Nov 17
Read Section 6.3

We discussed Principle of Inclusion and Exclusion following this handout. We solved one of the problems in class. We proved the Principle of Incl/Excl by induction on the number of sets.

Nov 19
(1) We gave a second proof of the principle of inclusion and exclusion.
(2) The proof in (1) relied on the statement that the alternating sum of the binomial coefficients on a single row of Pascal's triangle is zero. We proved this fact, and noted that it implies that half of subsets of a $k$-element set have even cardinality and half have odd cardinality.
(3) We used inclusion/exclusion to show that the number of surjective functions from an $n$-element set to a $k$-element set is $$ \sum_{i=1}^k (-1)^i\binom{k}{i}(k-i)^n $$
(4) We defined the Stirling numbers of the Second Kind and proved that $S(n,k)$ is the number of partitions of $n$ into $k$-cells.
(5) We noted that a surjective function is determined by its coimage and its induced map, and from this it follows that $$ S(n,k)=\frac{1}{k!}\sum_{i=1}^k (-1)^i\binom{k}{i}(k-i)^n $$
(6) We started discussing the recursive definition of $S(n,k)$.
Nov 29
We discussed the recursive definition of $S(n,k)$ and the Bell numbers following these slides.

Quiz 10 is at 3-10pm on Canvas! Last One!

Final Exam Review Sheet!

Dec 1
Discrete probability!

Read Section 6.10.

(Test yourself on the terminology for discrete probability with this Quizlet link: https://quizlet.com/646973631/terminology-for-discrete-probability-flash-cards/?x=1qqt.)

Final Exam Review Sheet!

Dec 3
We reviewed terminology about discrete probability, and calculated the probabilities for each type of poker hand.

Final Exam Review Sheet!

Dec 6
We reviewed terminology about discrete probability, and calculated the probabilities for a sequence of flips of an unfair coin. We also discussed the problem that arises if one tries to assign a probability to every subset of an infinite sample space (Banach-Tarski Paradox).

Final Exam Review Sheet!

Dec 8
We discussed the Final Exam Review Sheet. (The final exam will be Saturday, December 11, 4:30-7pm in ECCR 108.)

Final Exam Review Sheet!