Dr. Timo de Wolff
Institute of Mathematics
www.math.tamu.edu/~dewolff/Spring16/math302.html
MATH 302 – Discrete Mathematics – Section 501
Final Exam Review Homework
Spring 2016
Exercise 1. We denote all even integers by 2Z. Let for every j ∈ Z Ij denote the closed
interval of real numbers from j/2 to j/2 + 1. For example, I5 = [2.5, 3.5]. Compute the
following sets:
S
1. j∈2Z (Ij ∩ Ij+1 )
S
2. j∈2Z (Ij \ Ij+1)
S
3. j∈2Z (Ij ∩ Ij+1 ) ∩ (Ij \ Ij+1 )
S
S
4.
I
∩
I
j
j
j∈2Z
j∈Z\2Z
Exercise 2. Solve the following counting problems:
1. In how many ways can five distinguishable hand holds be distributed to eight (distinguishable) spots on a climbing wall?
2. In how many ways can one order a green, a blue, and two black balls?
3. In how many ways can one form a (single) team of 4 persons out of 10 persons?
4. How many lists with 10 entries exist if every odd entry is a lowercase letter and every
even entry a digit 0, 1, . . . , 9?
5. Recall the lizards in our garden (homework 8, exercise 3). Recently one of the lizards
living under the house visited the two in the old tree and changed his color to brown,
when he did. The other two lizards in the old tree decided since then not to take
brown color. If the other assumptions from the exercise remain valid: how many
combinations of colors can the 5 lizards have now?
6. Assume a group of 10 distinguishable persons vote for their favorite movie from a list
of movies A, B, C, D. Assume the outcome is (4, 2, 1, 3). How many combinations of
votes are possible, which turn into this result?
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7. How many different strings can be made from the letters in MISSISSIPPI, using all
the letters?
Exercise 3. Solve the following linear recurrence relation:
an = −5an−1 − 6an−2
with initial conditions a0 = 2, a1 = 1.
Exercise 4. Assume that containers with the amount of k ∈ N∗ TEU (twenty-foot equivalent units) were distributed to eleven ships at the Houston harbor on the last weekend. How
large has k to be such that we can guarantee that at least one ships carries containers of at
least 5001 TEU? Prove your answer.
Exercise 5. Prove by mathematical induction: 3 divides n3 + 2n for all n ∈ N∗ .
Exercise 6. Show the following equation for all n ∈ N∗ :
3
n
n X
n j
2
=
j
j=0
Exercise 7. Assume that three sets A, B, C satisfy
#A = #B = #C = 10, #(A ∪ B ∪ C) = 21, #(A ∩ B) = 5 and A ∩ C = ∅.
What is the cardinality of B ∩ C?
Exercise 8. Assume that two algorithms represented by functions f and g can be applied
to the same problem. Assume furthermore that the number of steps for a fixed input size s
is given by f (1) = 1 and
g(s) = s3 , and
f (s) = 4 · f (s/2) + 27s2
for all s ∈ {2, 4, 8, ...}. Which of the algorithms f or g will you choose for very large s?
Explain your answer.
Exercise 9.
1. Prove that the relation ∼ on R given by
∀a, b ∈ R : a ∼ b :⇔ ⌈a⌉ = ⌈b⌉
is an equivalence relation.
2. Prove that every equivalence class of ∼ has the same cardinality.
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3. Let {2}∪{3}∪· · · ∪{12} be a partition of the set {2, . . . , 12}. Show that this partition
is given by an equivalence relation, which can be defined by two dice. Compute the
cardinalities of each equivalence set.
4. Let R be a relation on R2 (i.e., it is a subset of R2 × R2 ) such that for all pairs (a, b)
and (c, d) it holds:
√
√
a2 + b2 ≥ c2 + d2 .
(a, b)R(c, d) :⇔
Show that this relation is reflexive and transitive. Is it also antisymmetric (and hence
a partial ordering)? How could the relation be changed slightly in order to make it
asymmetric (this would prevent it from being reflexive anymore, of course)?
Exercise 10.
1. Let A, B be two countable infinite sets and C, D two uncountable infinite sets. Assume
you are allowed to construct a new sets out of A, B, C, D via using two intersections,
one union and arbitrary many brackets. Will the resulting sets always be countable,
uncountable or is both possible? Explain your answer.
2. Prove that Z2 (i.e., Z × Z) is countable.
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