Math 100
FINAL EXAM
3/19/2009 , Dr. Frank Bäuerle, UCSC
NAME:
Note: Show your work.
Problem 1:
out of 20
Problem 2:
out of 20
Problem 3:
out of 20
Problem 4:
out of 20
Problem 5:
out of 20
Problem 6:
out of 20
Problem 7:
out of 20
Problem 8:
out of 20
Total:
out of 160
Good luck and have a good spring break!
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1. (20 points)
(a) Determine (with proof) whether the following propositional formula is a tautology,
contradiction or neither.
((P −→ Q) −→ R) ←→ (P −→ (Q −→ R))
(b) For this part of the problem, let the universe be the set of real numbers. Assume
that f is a real-valued function (i.e. domain and codomain of f are real numbers).
Describe what the following propositional formula asserts about the function f .
Then either prove it or give a counterexample to disprove it.
∀x ∀y (x < y −→ f (x) < f (y)) −→ ∀x ∀y (f (x) = f (y) −→ x = y)]
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2. (20 points) Recall that a relation ∼ is an equivalence relation on a set S provided that
∼ is reflexive, symmetric and transitive.
(a) Define precisely what it means for a relation ∼ on a set S to be
i. reflexive
ii. symmetric
iii. transitive
(b) Now let Z be the set of integers and define a relation ∼ on Z as follows:
For n, m ∈ Z define n ∼ m if there is a k ∈ Z such that n + m = 2k.
i. Give three different members of [−17] .
ii. Prove that ∼ is an equivalence relation on Z.
iii. (Extra Credit, 5 points) Now define for n, m ∈ Z that n ∼2 m if there is a
k ∈ Z such that n + m = 3k. Is ∼2 reflexive, symmetric, transitive?
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3. (20 points) Use induction to prove the following statements:
(a) Let the Fibonacci sequence be given by a1 = a2 = 1 and an+1 = an + an−1 for all
n ∈ N, n ≥ 2. Prove that
n
!
a2k−1 = a2n
k=1
(b) Use strong induction to prove that every natural number greater than two is either
a prime or can be written as a product of primes.
4
4. (20 points) Under the assumption that every integer z is either even or odd prove that
an integer z is even iff z 2 is even.
5
5. (20 points) Let A, B be subsets of a given universal set U . Recall that we defined
A − B = {x | x ∈ A ∧ x ∈
/ B}. Either prove or disprove (by giving a counterexample)
the following statements:
(a)
A − B )= ∅ −→ B ⊆ A
(b)
B ⊆ A −→ A − B )= ∅
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6. (20 points) Let R be the set of real numbers. Let f : R −→ R be defined via
f (x) =
1
1 + x2
(a) Find the following images and inverse images of sets:
i. f (R)
ii. f ((0, ∞))
iii. f ((−1, 1))
iv. f ([0, 1])
v. f −1 (R)
vi. f −1 ({ 12 , 0})
(b) Find a non-empty set A ⊆ R such that f −1 (f (A)) = A
(c) Find a non-empty set B ⊆ R such that f (f −1 (B)) = B
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7. (20 points) These problems are of a combinatorial nature. Do one of the following two
for credit and the other one for extra credit (5 points).
(a) Nine chairs in a row are to be occupied by six students and the three Professors
Alpha, Beta and Gamma. These three professors arrive before the six students
and decide to choose their chairs so that each professor will be between two
students. In how many ways can the Professors Alpha, Beta, and Gamma choose
their chairs ? Justify your answer.
(b) Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first,
middle, and last initials) will be in alphabetical order with no letters repeated.
Assuming that there are a total of 26 distinct letters A, B, . . ., Z, how many such
monograms are possible? Justify your answer.
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8. (20 points) Recall that for sets A, B we defined that A and B have the same cardinality
if there is a one-to-one function f : A −→ B that maps A onto B.
(a) Show that the set of natural numbers including zero, N0 = {0, 1, 2, 3, . . .} and the
set of natural numbers without zero, N = {1, 2, 3, . . .}, have the same cardinality.
(b) Show that the set of natural numbers including zero, N0 = {0, 1, 2, 3, . . .} and the
set of integers, Z, have the same cardinality.
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