- Assignment #1 (Assigned 3 Sept, due in class 5 September):
E&C define a grid game played by two players on a four by four
grid of squares. All squares are initially blank. On her move, a player
chooses one row or one column and places X's in one or more blank squares
in that row or column. (She may not choose a column that is all X's.) The
winner of the game is the player who places the last X, that is, the first
player who cannot move on her term loses.
A hard problem would be to determine who wins the four by four game if
both players play optimally. Your assignment is to determine who wins the
two by two game, and justify your conclusion by explaining how that player
can win the game no matter what their opponent does. If you succeed in this,
take a look at the three by three game.
- Assignment #2 (Assigned 5 Sept, due in class 8 Sept):
Section 1.2 of the book is about integer sequences such as (1, 3, 5,
7, ...). If we call this sequence A, then A(1) = 1 (the first
element of the sequence), A(2) = 3, A(3) = 5, and so forth.
We can give a sequence by a list "(1, 3, 5, 7,...)" as I did above, or with a
closed
form like "A(n) = 2n - 1", or with a recursive definition
like "A(1) = 1, and for any i > 0, A(i+1) = A(i) + 2". An
equivalent
form of the same recursive definition is "A(1) = 1, and for any i
> 1, A(i) = A(i-1) + 2".
Here are your two exercises, #7 and #8 on page 21 of the book.
#7: For each of these sequences given in closed form, write out
the
first five (or more if necessary) terms of the sequence. Use your
answer
to discover a recursive formula for the sequence.
- (a) a(n) = 3n + 1
- (b) b(n) = 5n - 2
- (c) c(n) = 2n + 7
- (d) d(n) = 3^{n} + 1
- (e) e(n) = 7n - 6
- (f) f(n) = 3n^{2} + n
- (g) g(n) = 2^{2n-1} - 1
#8: For each of these sequences give recursively, write out the
first
five (or more if necessary) terms of the sequence. Use your
answer to discover a closed formula for the sequence.
- (a) a(1) = 5, a(n) = 5a(n-1)
- (b) b(1) = 2, b(n) = 5b(n-1)
- (c) c(1) = 3, c(n) = 2a(n-1)
- (d) d(1) = 4, d(n) = 4 + d(n-1)
- (e) e(1) = 1, e(n) = 4 + e(n-1)
- (f) f(1) = 6, f(n) = 3 + f(n-1)
- Assignment #3, due 10 September:
(E&C 1.2.25, adapted)
Last week I explained the Josephus game, which is best
explained now by a
web link to the authors' page. For a given number of people and
a given skip number, you can follow the procedure and see who is the last
person to survive. We say that J(p, s) number of that person. For
example, playing the game as it comes up on the site, we can see that J(6, 2)
= 5.
- (a) For each value of p from 2 through 12, determine J(p, 2).
- (b) If you did this correctly, there should be a pattern in the answers
that you can figure out. Describe this pattern in words and use it to
predict the answer for 13, 14, 15, and 16 people, still with a skip of 2.
- (c) Predict the answer for 31, 42, and 53 people. Note that the web
site can't handle more than 30 people and just does it for 30 if you give it
a higher number.
- Assignment #4, due 12 Sept:
In the book, Section 1.3 exercise 11 parts b, c, and e only (since a
and d are in the back of the book), exercise 21 parts b, d, and e,
exercise 22.
- Assignment #5, due 15 Sept:
In the book, Section 1.3 exercises 7 (parts b and d only) and 18,
and Section 1.4 exercises 2, 9, and 16 (a and c only).
- Assignment #6, due 17 Sept:
In the book, Section 1.4 exercises 19 and 20, and Section
1.5 exercises 1 (parts b, d, f, g only) and 4 (parts b, d, f only).
- Assignment #7, due 19 Sept:
In the book, Section 1.5 exercises 17, 18,
28 (part b only), 30 (parts b, d, f, g
only), and Section 1.6 exercises 5 and 7.
- Assignment #8 due 22 Sept:
E&C 1.3.9: Use the letter s to represent the statement
"Chris likes to play soccer", r for "Chris likes to read",
amd p for "Chris likes to eat pizza". Use these names along
with the basic logic operations (AND, OR, and NOT) to write
each of following English sentences in symbolic logic notation:
- (a) Chris likes pizza but he does not like soccer.
- (b) Chris likes to read and eat pizza, or he likes to play
soccer.
- (c) Chris does not like to eat pizza, but he likes to play
soccer or read.
- (d) Chris likes to do two of these things but not all three.
E&C 1.3.23 parts c and d only: Use the double negative property
and DeMorgan's laws to rewrite each of the following as an
equivalent
statement that never has the NOT symbol ¬ outside of a
parenthesized expression. The first one is done for you as an
example. (The two DeMorgan laws say that ¬(x ∧ y)
is equivalent to ¬x ∨ ¬y and that ¬(x ∨ y) is
equivalent to ¬x ∧ ¬y. The Double Negative Law
says that ¬(¬x) is equivalent to x.)
- (ex) ¬(p ∧ ¬q) is equivalent to ¬p ∨
¬(¬q) by DeMorgan's law, and this is equivalent to ¬p
∨ q by the double negative property.
- (c) ¬(¬(¬p ∧ q))
- (d) ¬(p ∧ (q ∨ ¬ p))
E&C 1.6.7: Use truth tables to characterize each of the following
propositions as a tautology (always true), a contradiction (always
false), or neither:
- (p → (q ∧ r)) ∨ ((p ∧ q) → r)
- ((p → q) ∧ (q → ¬p)) → ¬p
- ((p → q) ∧ (¬p → r)) → (q ∨ r)
- Assignment #9, due 24 September: The worksheet "A Murder
Mystery", sent by email.
- Assignment #10, due 26 September: The worksheet
"Translating
Predicates", handed out on paper. If possible, also look at the
worksheet "Translating Quantifiers".
- Assignment #11, due 29 September: Complete the
worksheet
"Translating Quantifiers" if you can (definitely do parts 3 and 4). Also do the following
translation exercises, with variables of type "dog", predicates
T(x) for "dog x is a terrier" and S(x, y) for "dog x is sillier
than dog y", and constant dogs c
"Cardie" and d "Duncan". You will also need the "=" operator in
symbolic statements.
- "Every dog is sillier than some terrier who is siller than
Cardie."
- "There is a dog who is siller than Cardie but not sillier
than any dog who is not Cardie or Duncan."
- "For every dog, there is a different dog who is sillier
than both the first dog and Duncan."
- ∃x:∀y:∃z:S(x, y) → (T(z) ∧
S(y, z))
- ∀x:∀y:∃z:S(x, y) → (¬T(z)
∧ S(x, z) ∧ S(z, y))
- ∃x:∀y:(S(y, c) ∧ S(y, d)) → ((y = c)
∨ (y = d))
- Assignment #12, due in class 3 October 2014:
First two more translations like those in Assignment 11:
- "Cardie is siller than every terrier except Duncan."
- ∀x:∃y:T(y)∧(∀z:S(x, z) →
S(y, z))
Then in the book, do Exercises 2.1.3 and 2.1.10 on pages 96
and
97. Remember that exercises in blue are done in the back, so
that you can get hints about 2.1.3 from the solutions to the
similar 2.1.2, for example.
- Assignment #13, due in class 6 October 2014:
First, two more English-to-symbols translations. The first
has variables of type "dog" and predicates as in Assignments
11 and 12. The second has predicates of type "positive
integer"
and predicates E(x) for "x is even" and O(x) for "x is odd".
- "Every terrier is either sillier than some non-terrier
or sillier than Cardie."
- "There is an odd number between any two different even
numbers.
In the book, do Exercises 2.1.4 (parts b, d, f only),
2.1.12 (parts b, d, e only), 2.2.4, and 2.2.10.
- Assignment #14, due in class 8 October 2014:
In addition to discussing Assignment #13 in class, we will
look at the examples in Section 2.2 and the first example in
Section
2.3. Write up Exercises 2.2.15 and 2.2.17.
- Assignment #15, due in class 10 October 2014:
Along with 2.2.15 and 2.2.17, write up 2.2.14 (b) and 2.3.2.
- Assignment #16, due in class 14 October 2014:
Exercises 2.3.3 (parts b, d, and f only), 2.3.4, and 2.3.5.
- Assignment #17, due in class 15 October 2014:
Exercises 2.3.9b, 2.3.13, and the following:
"Prove by induction that the sequence defined by the rules a_{1} = 1 and
a_{m} = a_{m-1} + m^{2} satisfies the closed form
a_{n} = n(n+1)(2n+1)/6."
- Assignment #18, due in class Wed 22 October
2014:
Exercises 2.5.2, 2.5.12, 2.5.14 parts b and c.
- Assignment #19, due in class Fri 24 October 2014:
Exercises 3.1.2, 3.1.6, 3.1.12 (parts b, d, e), 3.1.17 (parts
a, b, d, f, g), 3.1.28, 3.1.30.
- Assignment #20, due in class Mon 27 October
2014:
Exercises 3.1.18, 3.1.25 (you are asked to find a formula
for m(A ∪ B ∪ C ∪ D)), 3.2.10 (parts b and d),
3.2.20, 3.3.2 (parts b, d, e), and 3.3.10.
- Assignment #21, due in class Wed 29 October 2014:
Do the remaining problems from last time (3.2.20, 3.3.2 bde,
3.3.10), plus 3.2.18, 3.3.6bc, and 3.3.13bc.
- Assignment #22, due in class Fri 31 October 2014:
Exercises 4.1.1 (parts b and d),
4.1.5, 4.1.9, 4.1.11, 4.1.15 (this is a multiple
choice question, not a question with parts -- explain your
answer), 4.1.18.
- I forgot to post an assignment due Mon 3 November, sorry.
- Assignment #23, due in class Wed 5 November:
Exercises 4.2.2, 4.2.7, 4.3.2 part (b) only, 4.3.4 parts (b) and (c),
4.3.12, 4.3.20, 4.3.22.
- Assignment #24, due in class Fri 7 November:
Exercises 5.1.2, 5.1.4 (parts a, b, d, f), 5.1.8, 5.1.12
(parts a, b, d, f), 5.1.13 (parts b, d), 5.1.19.
- Assignment #25, due in class Mon 10 November:
(Read Section 5.2, which we would have gone over in class
today if either of you were there.) Exercises 5.2.2, 5.2.4,
5.2.10, 5.2.15.
- Assignment #26, due in class Fri 14 November:
(Read Section 5.3) Exercises 5.3.2 (b, d), 5.3.4, 5.3.9,
5.3.18, 5.3.20, 5.3.22
- Assignment #27, due in class Mon 17 November:
(Read Section 5.4) Exercises 5.4.2, 5.4.4, 5.4.8, 5.4.12,
5.4.18, 5.4.20.
- Assignment #28, due in class Wed 19 November:
(Read Sections 6.1, 6.2) Exercises 6.1.2 (b, c), 6.1.6,
6.1.10, 6.1.20, 6.2.2 (b, c), 6.2.6.
- Assignment #29, due in class Fri 21 November:
(Read Section 6.3) Exercises 6.2.14, 6.2.18, 6.3.2, 6.3.6,
6.3.10, 6.3.15.
- Assignment #30, due in class Mon 24 November:
(Read Section 6.4) Exercises 6.3.19 (parts a, b), 6.4.2,
6.4.4, 6.4.7, 6.4.10, 6.4.20.
- Assignment #31, due in class Mon 1 December:
(Read Appendix B) Review the exercises for Appendix B, which
have solutions in the text. Then consider the matrix
1/2 1/4 1/4
A = 1/4 1/4 1/2
1/2 1/4 1/4
and compute the matrices A^2 (meaning "A times A"), A^3, A^4, and A^{10}.
- Assignment #32, due in class Tue 3 December:
(Read Section 6.6, pp. 482-486) Exercises 6.6.2, 6.6.4, 6.6.6, 6.6.8,
6.6.10. If you do not have the appropriate dice available, you can simulate them using the random integer generator
here.