Assignment
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Assigned
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Due
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Problems
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| HW1 |
8/26/09
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9/2/09
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Assignment
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HW2
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9/2/09
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9/11/09
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For the following problems, you may use any rules of arithmetic you learned in some class.
1. Show by induction that 1 + 3 + 5 + ... + (2n-1) = n2.
2. Show by induction that that n < 2n.
3. Show by induction that 1 + 1/√2 + 1/√3 + ...
+ 1/√n < 2*√n.
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HW3
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9/10/09
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9/16/09
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For Problems 1 and 2 you may use only those
rules of arithmetic we proved in class. (If you need some other rule,
then you must prove it first.)
1. Prove the following laws of arithmetic:
(a) The associative law for multiplication.
(b) The commutative law for multiplication.
2. Prove that (m*n)k = (mk)*(nk).
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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HW4
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9/16/09
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9/23/09
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1.
Write (p ∨ q ∨ r) in disjunctive normal form.
2.
Give a direct proof, a proof of the contrapositive,
and a proof by contradiction of the following statement:
for any sets A, B and X, if
(X ∩ A) = (X ∩ B), and (X ∪ A) = (X ∪ B), then A = B.
3.
Suppose you want to prove a theorem with two hypotheses:
((H1 ∧ H2) ⇒ C). Which
of the following proof strategies would suffice
to prove the theorem? Explain your answer.
(a) Prove
(((¬C) ∧ H2) ⇒ (¬H1)).
(b) Prove
((¬H1) ∨ (¬H2) ∨ C).
(c) Prove
((¬C) ⇒ ((¬H1) ∧ (¬H2))).
(Hint: (c) is a little tricky.)
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HW5
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9/23/09
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9/30/09
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1.
(a) Write the pairing axiom as a formal sentence.
(b) Write the power set axiom as a formal sentence.
2. 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 and false in the real numbers.
(c) true in the real numbers and false in the natural numbers.
(In this language the only nonlogical symbol is "<".)
3.
Is
∀ w ∃ x ∀ y &exist z
(w2 + x2 = (y + z)2)
true or false in the real numbers? Support your answer by
giving a winning strategy for the appropriate quantifier.
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HW6
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9/30/09
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10/7/09
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1.
(a)
Write ((∀ x P(x))⇒(∃ x P(x))) in prenex form.
(b) Use a quantifier game to prove that the sentence in (a)
is true in any structure that has a predicate symbol P.
2.
(a)
Let F(x) =
♦ +1 if x is a positive real number
♦ 0 if x = 0
♦ -1 if x is a negative real number
Use a quantifier game to
show that F is not continuous at x=0.
(b) Let G(x) =
♦ 1/x if x is not zero
♦ 0 if x = 0
Use a quantifier game to
show that G is not continuous at x=0.
3.
Do Problem 1 from Chapter 4.
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HW7
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10/7/09
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10/14/09
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1. 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.
2. Prove or disprove:
(a) The union of two equivalence relations is an equivalence
relation.
(b) The intersection of two equivalence relations is an equivalence relation.
3. Let S be the set of all propositions that can be formed
using some nonempty set of propositional variables.
(a) Show that ``equivalence of propositions'' is an equivalence
relation on S.
(b) Describe a function whose kernel is the equivalence relation
of part (a).
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10/14/09
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10/21/09
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No HW.
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HW8
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10/22/09
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10/28/09
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Read Section 5.3. Our notation differs slightly from
that of the book: we write [(a,b)]E for the
integer that the book writes
as [(b,a)].
1. Prove the following statement about the integers:
∀ x
∀ y
∀ z
(x(y+z) = xy + xz).
2. Prove the following statements about the integers.
(a) ∀ x ((-1)*x = -x).
(b) ∀ x
∀ y
((x < y) ⇒ (-y < -x)).
(c) The product of two positive integers is positive.
3. Let E be an equivalence relation on a set X. Under what
circumstances is the function
F: X/E → X: [x] mapsto x
well defined?
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10/28/09
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11/4/09
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No HW.
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HW9
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11/4/09
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11/11/09
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Read Sections 6.2, 6.3, 6.8.
1. How many ways are there to order {1, 2, ..., n} so that
these elements form two incomparable chains? (Example: If n=5,
then one chain might be 1<3<4 while the other chain might be 5<2.)
2.
(a) Find the coefficient of x2y3z4
in (x+y+z)9.
(b) Find the coefficient of x3y3z3
in (2x+3y+5z)9.
3. Find the coefficient of x15
in (2x+3x2+5x3)9.
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HW10
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11/11/09
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11/18/09
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1. Use inclusion/exclusion to determine how many numbers between 1 and 200 are
(a) not perfect n-th powers for any n>1.
(Not perfect squares, not perfect cubes, etc.)
(b) square free. (That is, not divisible by n^2 for any n>1.)
(Hint for (b): 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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HW11
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11/18/09
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12/2/09
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Read Section 6.7.
1.
(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?
2. How many ways are there to arrange a deck of 52
cards with no adjacent hearts?
3. If you roll a fair die three times, what is the probability
that the results are in nondecreasing order?
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HW12
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12/3/09
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12/9/09
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Read Sections 8.1-8.3.
1.
A dodecahedron is a 12-sided solid whose faces are pentagons.
(See picture.)
Give a planar drawing of the graph whose vertices are
the vertices of the dodecahedron (the points where 3 faces meet)
and whose edges are the edges of the dodecahedron (the segments
where 2 faces meet).
2.
What is the probability that a loopless graph with n vertices
has no isolated vertices? (A vertex is isolated if it
is incident to no edge.)
3.
Which complete bipartite graphs Km,n are planar?
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