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Math 2001: Discrete Mathematics,
Spring 2011
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Homework
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Assignment
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Assigned
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Due
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Problems
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| HW1 |
1/12/11
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1/19/11
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Read Section 9.
Assignment
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| HW2 |
1/19/11
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1/26/11
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Read Sections 1-6.
1. Prove or disprove:
(i) P(A∩B) = P(A)∩P(B).
(ii) P(A∪B) = P(A)∪P(B).
(Here P = power set.)
2. Prove or disprove: for any A, B and X,
if A∪X = B∪X and A∩X = B∩X,
then A=B.
3. Show that A⊆B if and only if A∩B=A.
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| HW3 |
1/27/11
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2/2/11
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Read Section 19 (proof by contradiction)
and Section 10 (quantifiers).
1. Write the following axioms of set theory as formal sentences.
(i) Pairing.
(ii) Power set.
2. In 1959, Pete Seeger took lines from
the Book of Ecclesiastes to write a song,
which was made famous by the Byrds in 1965. One line is:
To every thing there is a season,
and a time for every purpose under heaven.
Write this as a formal sentence using predicates
S(s,t) = "s is the season for thing t" and
P(T,p) = "T is the time for purpose p".
3. Write a formula Prime(x) that is true for a positive integer x
if it is prime and is false for all other positive integers.
Use whatever standard symbols you need.
For extra practice with problems like these, try Exercises 10.1(a)-(k).
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| HW4 |
2/4/11
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2/10/11
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1.
Is
∀ w ∃ x ∀ y ∃ 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.
2.
(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 nonempty structure that has a predicate symbol P.
3.
Let G(x) =
♦ 1/x if x is not zero
♦ 0 if x = 0
Use quantifier games to
show that G is not continuous at x=0, but is continuous at x=1.
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| HW5 |
2/9/11
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2/16/11
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Read Section 21 (Induction).
1.
Show that
1+1/√2+1/√3+…+1/√n<2*√n.
2. Do parts (b) and (d) of Exercise 21.8 from the book.
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. At each stage,
all pieces should be rectangles whose sides have integer length.)
For extra practice with problems like these, try Exercises 21.3, 21.4,
21.5, 21.9, 21.15.
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| HW6 |
2/16/11
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2/23/11
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Read Sections 23 (Functions) and 16 (Binomial Coefficients).
1. Do Exercise 23.5 of the text.
2. Show that |A|≠|P(A)| for any set A. (Hint:
Prove that no function from A to P(A) can be onto.
Start by assuming that f:A→P(A) is onto. Define
U={a∈A | a∉f(a)}, which is an element of P(A).
Show that there can be no u∈A such that
f(u)=U by considering whether
u∈f(u) and u∈U. Contradiction!)
3. This problem involves a deck of 52 distinct playing cards.
(a) In how many ways can a 13 card bridge hand be dealt from the deck?
(b) How many different 13 card bridge hands are there?
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| No HW |
2/23/11
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3/2/11
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No HW.
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| HW 7 |
3/2/11
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3/9/11
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1. Do Exercise 16.11 of the text.
2. Do Exercise 16.28 of the text.
3. Find the coefficient of x15
in (2x+3x2+5x3)9.
For extra practice, try 16.3, 16.7, 16.8, 16.14, 16.15.
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| HW 8 |
3/9/11
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3/16/11
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1.
(a) How many paths are there from the point (0,0)
of ℝ2
to the point (10,15) of ℝ2 if each path
consists of a sequence of steps of length 1 moving
in the direction of the positive x-axis or the positive y-axis?
(b) How many paths are there from the point (0,0,0)
of ℝ3
to the point (10,15,20) of ℝ3 if each path
consists of a sequence of steps of length 1 moving
in the direction of the positive x-axis, the positive y-axis
or the positive z-axis?
2. Let MC(n;k) be the number ``n-multichoose-k''. Use a combinatorial
argument to show that
MC(n;0)+MC(n;1)+…+MC(n;k)=MC(n+1;k)
3. How many ways are there to make 3 fruit baskets from 8 pineapples,
10 pomegranates, 6 coconuts and 20 figs if each basket must contain
each kind of fruit?
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| HW 9 |
3/16/11
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3/30/11
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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 5-card poker hands have cards of every suit?
3. How many solutions in ℕ are there to
x1 + x2 + x3 + x4 + x5
= 50
if
8≤x1≤13,
6≤x2≤15,
6≤x3≤14,
9≤x4≤14, and
7≤x5≤16?
For extra practice, try Exercises 18.1, 18.3, 18.5 and
Chap 3 Self Test 19 and 20.
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| HW 10 |
3/30/11
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4/6/11
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1.
(a) Let f:ℝ→ℝ:x↦x2
be the squaring function from the reals to the reals.
Describe
(i) dom(f)
(ii) cod(f)
(iii) im(f)
(iv) coim(f)
(v) inc:im(f)→cod(f)
(vi) nat:dom(f)→coim(f)
(vii) f:coim(f)→im(f)
(b) Do the same for
g:ℝ→ℝ:x↦x4. Write just
"same" for an item (i)-(vii)
if the answer is the same as in (a).
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 or disprove:
(a) The union of two equivalence relations is an equivalence relation.
(b) The intersection of two equivalence relations is an equivalence relation.
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| HW 11 |
4/7/11
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4/13/11
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1.
(a) Recall that we defined ℤ using the intuition that
a pair (m,n) represents the "difference" m-n.
Show that this intuition is correct
by establishing that
[(m,n)]E =
mℤ+(-nℤ).
(b) Show that x(y+z)=xy+xz in ℤ.
2. The rational numbers are constructed as equivalence classes
of pairs of integers. Our intuition is that a pair
(p,q) represents the "ratio" p/q. Let
S = {(p,q)∈ℤ×ℤ|q≠0}, and define a relation
E on S by
((p,q),(r,s))∈E ↔ ps=rq.
Show that E is an equivalence relation on S. (The rational numbers
are defined on the set S/E.)
3. Let f:A→B be a function. Show that
f:coim(f)→im(f)
is well-defined.
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| HW 12 |
4/13/11
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4/20/11
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1. Do Exercise 46.14 of the text.
2. Do Exercise 47.7 of the text.
3.Write formal sentences expressing the following properties of graphs.
(a) Every vertex has degree three. (G is a 3-regular graph.)
(b) The independence number is at least three.
(c) The diameter is at least three.
Try these practice problems: 46.2, 46.13, 46.17, 47.4, 48.5, 48.7, 48.9.
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