Record of daily activities and homework,
Math 220, Fundamentals of Discrete
Mathematics, Honors Section 200,
Fall 2003
- Monday, September 1
- We discussed the notions of statements, open
sentences, universal and existential quantifiers, and
negation.
- Homework for Wednesday: Read section 1.1
(pages 1-12) and section 1.2 (pages 16-24), and do
exercise 3 on page 13 and exercise D1 on
page 14. (These two problems are not to be handed
in on Wednesday but rather to be discussed in class.)
- Wednesday, September 3
- We discussed negation in relation to the homework
exercise, and we discussed different approaches to
exercise D1 about number theory.
- Homework for Friday: Read section 1.3
(pages 29-35) and section 1.4 (pages 38-44), write up
your solution to exercise D1 on page 14 to
turn in, and do exercise 9 on page 36
(section 1.3) and
exercise 1 on page 44 (section 1.4) to turn in.
- Friday, September 5
- We continued the discussion of logical connectives,
in particular, the notions of necessary conditions and
of sufficient conditions. We worked out that the
biconditional and the exclusive or are both
associative.
- Homework for Monday: Do exercise 8 on
page 13 (section 1.1), exercise 8 on
page 26 (section 1.2), exercise 12 on
page 36 (section 1.3), and exercise D1 on
page 46 (section 1.4).
- Monday, September 8
- We looked at some examples of solving logical
puzzles involving multiple statements.
- Homework for Wednesday: Read
section 5.1, pages 151-155. Do exercise 1(b)
on page 157 (section 5.1) and exercise D7
on page 15 (section 1.1), and revise your
proof for exercise D1 on page 14
(section 1.1). Also do (for class discussion, not
to hand in) exercise D7 on page 47
(section 1.4).
- Wednesday, September 10
- We observed that existential and universal
quantifiers do not commute, and we looked at an axiom
system for the integers.
- Homework for Friday: Read pages 159-164 in
section 5.2. Do exercise 1b on page 169
(section 5.2) and exercises 7 and D1 on pages
157-158 (section 5.1).
- Friday, September 12
- We discussed the method of mathematical induction
and its connection with the well-ordering principle.
- Homework for Monday: Complete
exercise 4(a) on page 169 (section 5.2),
which we started in class, and read the short story
The Bottle Imp by Robert Louis Stevenson.
- Monday, September 15
- We compared induction with strong induction and
discussed the downward induction in The Bottle
Imp.
- Homework for Wednesday: Read about the
binomial theorem on pages 165-167 and Bernoulli numbers
on pages 167-169 and do exercises 4(b), 7, and 24 on
pages 169-171 (section 5.2).
- Wednesday, September 17
- We discussed binomial coefficients and Pascal's
triangle.
- Homework for Friday: Read section 2.1
(pages 49-57). Do exercises 1(g), 1(h), 3(a), 3(b),
and 8(d) on pages 57-58 (section 2.1) and exercise 37 on
page 172 (section 5.2).
- Friday, September 19
- In the context of the conjectures in the homework,
we discussed how to prove that two sets are equal. The
conjecture that the set of odd primes of the form
4k+1 is the same as the set of odd primes that can
be written as the sum of two squares is true, and you
know how to prove half of it; proving the other half
requires some number theory that you do not yet know.
The other conjecture, that every even number larger
than 5 can be expressed as the sum of two odd
primes is a famous unsolved problem of mathematics
known as the Goldbach
Conjecture.
- Homework for Monday: Read
section 2.2, pages 61-68, and do exercises 1(h),
2(f), and 17 on pages 68-70 (section 2.2).
- Monday, September 22
- In preparation for the examination,
we listed the main topics covered in the course so far.
We looked at uses and misuses of notation. We started a
proof of the irrationality of the square root of 2
based on the well-ordering principle: namely, if
a is the smallest positive integer that can be
the numerator of a ratio a/b of integers
representing the square root of 2 (supposing that some
such representation were to exist), then the claim is
that (2b-a)/(a-b) is another representation of
the square root of 2, and with a smaller positive
numerator; contradiction.
- Homework for Wednesday: Read
section 2.3 (pages 72-78), and for class
discussion do exercises 26 and D1 on page 80
(Chapter 2) and verify the claim in the above
proof about the irrationality of the square root
of 2.
- Wednesday, September 24
- We discussed the homework problems about the
cardinality of power sets and about the pigeonhole
principle.
- Homework for Friday: Study for the examination.
- Friday, September 26
- First examination
- Monday, September 29
- The graded examinations were returned. As an
application of the pigeonhole principle, we looked at a
proof that for every irrational number x,
there exist infinitely many rational numbers
p/k such that
|x-(p/k)| <
1/k2.
- Homework for Wednesday: Read
section 3.1 (pages 81-93) and do exercises 15, 17,
23, and D1 on pages 95-96 (section 3.1).
- Wednesday, October 1
- We discussed the homework problem about the notion
of local maximum, and we began a discussion about the
notions of injectivity and surjectivity.
- Homework for Friday: Read section 3.2
(pages 97-105) and do exercises 8, 10, 17, and 20
on pages 106-108 (section 3.2).
- Friday, October 3
- We discussed the homework problems, and we worked out the
number of all functions (and also the number of all injective
functions) between two sets of finite cardinality.
- Homework for Monday: Read section 3.3
(pages 110-118) and do exercises 1(b), 3(b), and 7
on pages 118-120 (section 3.3) and
exercise 35 on page 172 (section 5.2).
- Monday, October 6
- We looked at examples of functions on the real line
that are bijective but not monotone (exercise D1
on page 121 in section 3.3); also examples of
functions that are montone but not
bijective. In addition, we began discussing binary operations.
- Homework for Wednesday: Read
section 4.1 (pages 123-134) and do exercises 1
and 6 on page 134 (section 4.1).
- Wednesday, October 8
- We worked on some interesting exercises from
section 4.1: numbers 14, 28, 31, 32, D1,
and D2 on pages 135-138.
- Homework for Friday: Each group should
write up its solutions. Also read the start of
section 4.2, pages 139-141.
- Friday, October 10
- We discussed relations and equivalence
relations.
- Homework for Monday: Read the rest of
section 4.2 (pages 141-147) and do exercises 1(d),
3, and 10 on pages 147-148 (section 4.2).
- Monday, October 13
- We discussed problems 14, 31, 32, and D1 from last
time (section 4.1).
- Homework for Wednesday: Do exercises 4, 7,
and 13 on pages 148-149 (section 4.2) to hand in and
exercise D6 on page 150 (section 4.2) to discuss in class.
- Wednesday, October 15
- Following up on a question from last time, we
discussed Hilbert's thirteenth problem. Then we
continued the discussion of equivalence classes.
- Homework for Friday: Read pages 192-193 in
section 5.5. Do exercises 8 and 12 on
page 197 (section 5.5) and exercise 12
on page 170 (section 5.2).
- Friday, October 17
- We discussed congruence modulo n.
- Homework for Monday: Read the first part
of section 5.3, pages 175-177. Do
exercise 16(a) on page 170 (section 5.2),
exercise 4(b) on page 180 (section 5.3),
and exercise 15(a) on page 197
(section 5.5).
- Monday, October 20
- We discussed the computation of greatest common
divisors via the division algorithm and the application
to finding multiplicative inverses of congruence
classes in Zn.
- Homework for Wednesday: Read the remainder
of section 5.3, pages 177-179, and do exercises
10(a) and D2 on pages 180-181 (section 5.3).
Also review for the examination to be given on Friday.
- Wednesday, October 22
- We discussed some previous homework problems and
reviewed for the examination to be given on
Friday.
- Friday, October 24
- Second examination
- Monday, October 27
- We discussed the solutions to the second examination, including the
extra credit problem.
- Homework for Wednesday: Read
section 5.4, pages 182-186. Find all Pythagorean
triples (x,y,z) of positive integers between
1 and 50 such that
x2+y2=z2.
- Wednesday, October 29
- We discussed Pythagorean triples, the rational
parametrization of the circle, and unique
factorization.
- Homework for Friday: Read the parts of
section 5.5 that you have not already read. Do
three (your choice) of the even-numbered problems in
section 5.4 (pages 186-187).
- Friday, October 31
- We looked at exercises 6, 10, and 14 in
section 5.4 (pages 186-187).
- Homework for Monday: Read section 5.6
(pages 200-206).
- Monday, November 3
- We discussed Fermat's little theorem and its proof
by induction.
- Homework for Wednesday: Read
section 6.1 (pages 209-218) and do exercises 1
and 4 on page 218 (section 6.1).
- Wednesday, November 5
- We discussed the phenomenon of infinite sets
admitting bijections with proper subsets.
- Homework for Friday: Read the first part
of section 6.2 (pages 220-223) and do
exercise 11(f) on page 219 (section 6.1)
and exercise 4 on page 227
(section 6.2).
- Friday, November 7
- We discussed countable and uncountable sets: in
particular, the rational numbers, the irrational
numbers, the algebraic numbers, and the transcendental
numbers. We saw that a randomly chosen real number is
transcendental with probability one.
- Homework for Monday: Read the rest of
section 6.2 (pages 223-227) and do exercise 3
on page 227 (section 6.2) and exercise 9
on page 218 (section 6.1).
- Monday, November 10
- We discussed cardinality of sets and the
Schroeder-Bernstein theorem.
- Homework for Wednesday: Read
section 6.3 (pages 229-233) and do exercises 7(h)
and 17 on page 228 (section 6.2).
- Wednesday, November 12
- We discussed Russell's paradox, the axiom of
choice, and issues in the foundations of set
theory.
- Homework for Friday: Reread pages 76-78
about the Cantor set. Do exercise 28 on
page 80 (section 2.3); also prove that the
Cantor set has the same cardinality as the set of real
numbers.
- Friday, November 14
- We discussed the Zermelo-Fraenkel axioms for set
theory and the equivalence of the axiom of choice with
the well-ordering principle.
- Homework for Monday: Do the exercises at
the end of the handout on the Zermelo-Fraenkel
axioms.
- Monday, November 17
- We discussed how to use the Zermelo-Fraenkel axioms
of set theory to establish the existence of Cartesian
products, and we proved that the Cartesian product of
the set of real numbers with itself is equinumerous
with the set of
real numbers.
- Homework for Wednesday: Revise the
solution to the previous homework, and review for the
examination over Chapters 5 and 6 to be given on Friday.
- Wednesday, November 19
- We reviewed for the examination on Chapters 5
and 6 to be given on
Friday.
- Homework for Friday: Prepare for the
examination.
- Friday, November 21
- Third examination
- Monday, November 24
- We discussed solutions to the third
examination.
For amusement over the Thanksgiving holiday, there is a
crossword puzzle.
- Wednesday, November 26
- There will be no class meeting this day. Enjoy the
Thanksgiving holiday. The next class meeting will be
Monday, December 1.
- Monday, December 1
- We discussed some topics in combinatorics: the
principle of inclusion-exclusion, the product rule, and
permutations.
- Homework for Wednesday: (1) How many
numbers between 1 and 1,000,000 (inclusive) are either
squares or cubes (or both)?
(2) How many possible radio call signs are there?
(The rule for broadcast radio station call signs in the
United States is that—ignoring historical
exceptions—they begin with either K or W and are
four letters long.)
- Wednesday, December 3
- We discussed counting problems involving
combinations.
- Homework for Friday: (1) If a fair
coin is flipped five times, what is the probability of
getting (a) exactly two heads; (b) at least
two heads?
(2) Prove that (2n)! is divisible by
(n!)2 for every positive
integer n.
- Friday, December 5
- We continued the discussion of combinations,
including probabilities of poker and bridge hands and
also coefficients of multinomial expansions.
- Monday, December 8
- This was our last class meeting for the semester.
We did the course evaluations and discussed another
counting problem. The final examination is scheduled
for Tuesday, December 8 from 3:30 to 5:30.