Assignment
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Assigned
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Due
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Problems
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| HW1 |
1/14/09
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1/23/09
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Assignment
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HW2
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1/21/09
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1/28/09
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Read
Section 1.2. For practice, do problems 3, 5, 6 and 9 from Section 1.2.
For the following problems, you may use any rules of arithmetic you
learned in some class.
1. Do problem 1.2.8.
2. Do problem 1.2.16.
3. You want to divide a piece of chocolate into 1"x1" square
pieces to distribute to a group of children. The starting piece is an
m"x n" rectangle. Show that it can be divided by breaking pieces of
chocolate a total of mn-1 times, but not with fewer breaks. (Each
"break" divides only one piece, and must be a straight line parallel to
one of the sides of the piece.)
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HW3
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1/29/09
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2/4/09
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Read
Section 1.5.
1. Do problem 1.5.20.
2.
Show by induction on m that
the following 2 statements are true for all natural numbers m and n:
(a) 0+m=m,
(b)
S(n)+m=S(n+m).
3. Show by induction that m+n=n+m
for every m and n.
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HW4
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2/5/09
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2/11/09
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Read
Section 1.1.
1. Show by induction that every positive integer is a sum of distinct
Fibonacci numbers.
2. Show that any sequence of integers satisfying the recurrence
a(n+1) = an - a(n-1)
+ ... + a(n-98)
- a(n-99)
+ a(n-100)
is bounded. (A sequence a1, a2, a3, ...
is bounded if there are fixed integers
A and B such that A < ai < B holds for all i.)
(Hint: try to find information about
the sum an+1 + an.)
3. Show that |A| is less or equal to |P(A)| even if A is infinite.
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HW5
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2/12/09
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2/18/09
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Read
Section 3.1. The following problems all involve multinomial coefficients.
1.
(a) Find the coefficient of x2y3z4
in (x+y+z)9
(b) Find the coefficient of x3y3z
3
in (2x+3y+5z)9
2. Find the coefficient of x15
in (2x+3x2+5x3)9
3. How many ordered partitions of an n-element set into
5 (possibly empty) cells? (The cells can be of arbitrary size.)
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HW6
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2/18/09
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2/25/09
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1. Do problems 3.1.20 and 3.1.22.
2. Describe a binary relation on some set that is:
(a) reflexive and symmetric but not transitive.
(b) reflexive and transitive but not symmetric.
(c) symmetric and transitive but not reflexive.
3. Prove that if x, y and z are integers, then
x*(y+z) = (x*y) + (x*z). You may use anything you
know about the arithmetic of natural numbers
to prove this.
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HW7
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2/25/09
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3/4/09
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Read
Sections 2.1 and 2.3.
1. Do problems 2.1.10 and 2.1.12.
2. Do problem 2.1.26, but use n=225 instead of n=45.
3.
Do Advanced Exercise 1 in Section 2.1. (Note: The left hand side of the identity should be x^b - 1 instead of x^b.)
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No HW
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3/4/09
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3/11/09
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Midterm 3/06/09
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HW8
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3/12/09
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3/18/09
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1. Do problem 2.3.20.
2. Do Advanced Exercise 2.3.2.
3. Find all solutions to the following congruences.
(a) 4x=16 (mod 24).
(b) 5x=16 (mod 24).
(c) 6x=16 (mod 24).
Express answers in the form x = a, b, c, ... (mod 24) for appropriate
a, b, c (etc).
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HW9
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3/19/09
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4/1/09
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1.
Use inclusion/exclusion to determine how many numbers between 1 and 200 are
(a) not divisible by 2, 3, 5 or 7.
(b) not perfect n-th powers for any n>1. (Not perfect squares, not perfect cubes, etc.)
(c) square free. (That is, not divisible by n^2 for any n>1.)
(Hint for (c): it is enough to consider elements not divisible by the square of any prime.)
2.
How many 20 digit decimal numbers are there in which the digits
1, 2, 3 all appear?
3.
How many 5-card poker hands have cards of every suit?
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HW10
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4/2/09
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4/8/09
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1.
If you roll a fair die three times, what is the probability that the
results are in nondecreasing order?
2.
(a) How many positive integral solutions are there to x1+x2+x3+x4+x5 < 50?
(b) How many positive integral solutions are there to x1+x2+x3+x4+x5+x6 = 50?
(c) Which answer is bigger and why?
3. How many ways are there to arrange
a deck of 52 cards with no adjacent hearts?
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HW11
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4/10/09
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4/15/09
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Read Section 5.3.
1. Do problems 5.3.6 and 5.3.10.
2. Do problems 5.3.23 and 5.3.24. (Justify your answers.)
3. Give a direct proof, a proof of the contrapositive,
and a proof by contradiction of the following statement:
if A, B and X are sets, (X intersect A) = (X intersect B),
and (X union A) = (X union B), then A = B.
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HW12
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4/16/09
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Read Section 6.3.
1.
(a) Write the empty set axiom as a formal sentence.
(b) Write the axiom of extensionality as a formal sentence.
2.
(a) Write the pairing axiom as a formal sentence.
(b) Write the power set axiom as a formal sentence.
3. Find a formal sentence in the language of ordered sets that is
(a) true in the natural numbers and false in the integers.
(b) true in the integers numbers and false in the real numbers.
(c) true in the real numbers and false in the natural numbers.
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