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Math 2001: Discrete Mathematics,
Fall 2012
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Lecture Topics
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Date
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What we discussed/How we spent our time
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Aug 27
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Syllabus. Text.
Informal definition of "set". Notation "x∈A".
The axioms.
We discussed the axiom of extensionality.
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Aug 29
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We discussed the axioms of empty set, pairing,
union (of two sets) and separation.
I defined "subset".
I defined the successor of a set.
I defined the finite ordinals. I gave the recursive definition
of addition. I proved that 2+2=4.
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Aug 31
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I gave the recursive definitions
of multiplication and exponentiation.
We reviewed the first five axioms and introduced
the axioms of power set and infinity.
For the axiom of infinity, we first defined
"inductive set". I defined the set of natural
numbers to be the set of all elements common to all
inductive sets. I gave the full axiom of union
(asserting the existence of a union of a family
of sets indexed by a set).
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Sep 5
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I posed two sample quiz questions.
(Define ℕ.
Find P(P(P(∅))).)
Intersection versus union:
why do we need an axiom for union,
but we don't need an axiom for intersection?
I explained why if I={i} is a set with
one element, then
⋃i∈I Ai = Ai =
⋂i∈I Ai.
I explained why if I=∅ is a set with
zero elements, then
⋃i∈I Ai = ∅ and
⋂i∈I Ai is not a set.
We discussed
Russell's paradox and proved that there is no set of all sets.
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Sep 7
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Axioms of replacement, choice and regularity.
Axiom of regularity guarantees that there
is no infinite descending chain
…
A3∈A2∈A1∈A0.
Some consequences are:
1. A∈A is prohibited.
2. Both A∈B and B∈A together are prohibited.
3. Every set is normal.
4. Russell's paradox is avoided.
5. The union of all sets is not a set.
Practice problems. (We solved 1, 2 and 4 in class.)
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Sep 10
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Logic:
Atomic and compound propositions.
Logical connectives. Truth tables.
Quiz 1.
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Sep 12
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More on truth tables.
Tautologies and contradictions.
Logical equivalence.
Some particular equivalences:
1.
(¬(¬P))≡P
2. ((P∨Q)∨R)≡(P∨(Q∨R)),
((P∧Q)∧R)≡(P∧(Q∧R))
3. (P∨Q)≡(Q∨P),
(P∧Q)≡(Q∧P)
4. (P→Q)≡((¬P)∨Q)
5. (P↔Q)≡((P→Q)∧(Q→P))
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Sep 14
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Definition of "monomial" and "disjunctive normal form".
Every proposition can be put
in disjunctive normal form.
The connectives ∧, ∨ and ¬ suffice
to generate any proposition (up to equivalence)
that has at least one propositional variable.
We worked on this handout.
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Sep 17
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Formula trees.
Introduction to structures, predicates and quantifiers.
Quiz 2.
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Sep 19
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Introduction to structures, predicates and quantifiers, part 2:
I gave the definitions of (and intuition for)
"structure", "language" and "term".
A handout.
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Sep 21
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I gave the definitions of "atomic formula"
and "formula". We practiced reading and writing
formulas.
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Sep 24
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I defined "scope of a quantifier", "bound variable",
"free variable" and "sentence". We started a discussion
about how to determine the truth of a sentence
in a structure. We mentioned the notation
Q↑G(φ,S) for
"quantifier Q has a winning strategy in the game
defined by φ in the structure S".
Handout.
Quiz 3.
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Sep 26
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Quantifier games. We worked on the exercises
from the second page of the Sep 24 handout.
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Sep 28
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Prenex form. Handout.
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Oct 1
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Restricted quantifiers.
Handout.
Quiz 4.
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Oct 3
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We worked on this handout.
We solved 1(a), 1(b), 2(b) and 2(d) in class.
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Oct 5
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Satisfaction, consequence and proof.
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Oct 8
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Valid sentence versus tautologies.
Definition of "formal proof" and
informal definition of "informal proof".
Notation and intuition for "rules of deduction".
Modus ponens and modus tollens.
I circulated the following
review sheet
for the next exam.
Quiz 5.
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Oct 10
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Review.
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Oct 12
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Exam.
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Oct 15
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No quiz. We discussed formal rules of proof.
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Oct 17
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We discussed this handout
about proof writing strategies.
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Oct 19
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We practiced proof writing.
Handout.
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Oct 22
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We describe the method of proof by induction (Section 4.3).
We proved that 1+3+5+…+(2n-1)=n2 by induction.
We proved that induction is a valid form of proof.
Quiz 6.
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Oct 24
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We used induction to prove some
rules of arithmetic.
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Oct 26
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Strong induction.
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Oct 29
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We stated the Recursion Theorem, which asserts
that a correct recursive definition
defines a unique function. We introduced the Fibonacci numbers
and proved that
(3/2)n≤
Fn+2≤
2n.
Quiz 7.
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Oct 31
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Relations. (Read about Cartesian products in Sect 1.4, then read
Sections 5.1 and 5.3.)
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Nov 2
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Functions. Handout.
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Nov 5
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Thm. (Images are subsets)
The image of a function with codomain B is a subset
of B. Conversely, any subset of B is the image of some
function with codomain B.
Quiz 8.
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Nov 7
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Thm. (Coimages are partitions)
The coimage of a function with domain A is a partition
of A. Conversely, any partition of A is the coimage of some
function with domain A.
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Nov 9
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Defined the adjectives: reflexive, irreflexive, symmetric, antisymmetric,
asymmetric, and transitive and gave examples.
Defined the nouns: equivalence relation,
partial order, strict partial order and gave examples.
Defined: equivalence class [a]E and quotient set
A/E (read "A modulo E"). Stated the theorem:
Thm. (Equivalence relations versus partitions)
Let A be a set. If E is an equivalence relation on A, then
A/E is a partition of A. Conversely, if P =
{Ai : i∈I}
is a partition of A, then EP:=⋃i∈I
(Ai)2 is an equivalence relation on A.
Moreover, the functions E↦A/E and P↦EP
are inverses of one another.
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Nov 12
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We proved part of the theorem from Friday (that if E is an equivalence
relation on A, then A/E is a partition of A).
Quiz 9.
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Nov 14
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We finished talking about the theorem from last Friday
(abouth the correspondence between equivalence relations on A
and partitions of A. We defined the kernel of a function
(ker(f)={(a,b)∈A2 : f(a)=f(b)}),
and explained why ker(f) is an equivalence relation,
why any equivalence relation is the kernel of a function,
and why ker(f) is the equivalence relation associated to coim(f).
We showed that the kernels of x↦x2 and
x↦|x| are the same.
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Nov 16
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We discussed what it means for a function to be
"well-defined". We discussed the construction of the rational
numbers from the integers and noted that
f:ℚ→ℤ:p/q↦p+q is not well-defined.
We discussed the construction of the integers
modulo m from the integers and noted that addition
and multiplication of integers modulo m is well-defined,
but exponentiation of integers modulo m is not
well-defined. (That is, a(mod m)b(mod m):=
ab(mod m) is not a well-defined operation.)
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Nov 26
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Counting 1:
We started discussing this handout.
Specifically, we discussed the definitions of
|A|=|B|, |A|≤|B|, |A|<|B|, finite, infinite, countable
and uncountable. We proved that ℕ and ℤ
are countable. We established that the power set of ℕ is uncountable
by proving Cantor's Theorem: For any set A, |A| < |P(A)|.
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Nov 28
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Counting 2:
Sum Rule and Product Rule.
Quiz 10.
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Nov 30
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Counting 3:
Practice problems.
The number of functions from
an m-element set to an n-element set is nm.
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Dec 3
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Counting 4:
The number of bijections between two different n-element sets
is n! This is also the number of permutations
of a single n-element set. (A permutation
of set A is a bijection from A to itself.) The number of ways to
linearly order an n-element set is also n!
The number of injective functions from an m-element set to
an n-element set is (n)m = P(n,m) = n!/(n-m)!
The number of m-elements subsets of an n-element set is C(n,m)=n!/(m!(n-m)!).
Quiz 11.
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Dec 5
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Counting 5:
We proved some claims made in the Dec 3 lecture, then
worked on
practice problems.
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Dec 7
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Counting 6:
We introduced the formula for counting multisets.
We discussed this handout,
which organizes many counting formulas.
(Solution key.)
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Dec 10
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Counting 7:
We proved the Binomial Theorem (using a combinatorial argument),
introduced Pascal's Triangle,
established the validity of the
recursive definition of the binomial coefficients
based on Pascal's Identity (using a combinatorial argument),
explained why Pascal's
Triangle is symmetric about the central vertical axis
(using a combinatorial argument),
and showed that the sum of the numbers on the nth
row is 2n (using a combinatorial argument). We took
Quiz 12.
Here is a review sheet for the final.
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Dec 12
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Counting 8:
Stirling numbers of the second kind.
We discussed the definition of the Stirling numbers
and compared these numbers with the binomial coefficients.
We described an explicit formula for Stirling numbers,
the generating function for Stirling numbers,
and the meaning of the nth row sum of the table of Stirling numbers.
We proved a recursion formula for Stirling numbers
(using a combinatorial argument) and proved
that the number of surjective functions from a k-element set
to an n-element set is S(k,n)*n!.
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Dec 14
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We reviewed for the final.
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