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The final exam for the discrete mathematics course (cs 173) offered by the department of computer science at the university of illinois at urbana-champaign during the spring 2004 semester. The exam consists of multiple choice, short answer, and long problem questions covering various topics in discrete mathematics such as logic, sets, functions, relations, graphs, and polynomial algebra.
Typology: Exams
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Final Exam
Friday, May 7, 2004
Print your name neatly in the space provided below, and sign in the area provided. Do not place your social security number anywhere on this exam.
Name:
Netid:
Lecture Section:
Signature:
This is a closed book and closed notes exam. Try to do ALL problems in this booklet. Read each question very carefully. If you think that there is an ambiguous statement, make an explicit assumption, and keep on solving using your assumption. If you run out of time, at least try to explain what you are trying to do.
You should have 14 pages. Page numbered 14 is draft paper. You may remove it but you have to turn in the draft paper at the end. You can also use the back sides of the pages as draft area of area for the solution, if needed.
Question Points Score Grader
1 6
2 6 3 6 4 6
5 6 6 6 7 6
8 6 9 6 10 6
11 10 12 10 13 10
14 10 15 10 16 10
17 15 18 15
Total 150
You must return all pages of this exam.
Which of the following functions has the slowest order of growth?
a)
b)
c)
d)
What is the running time of the following algorithm? (Choose all that apply.)
for i = 1 to n j := n; while j >= 1 j := j / 2;
a)
b)
c)
d)
Given that ^ ^ !^ , which of the following recursive definitions could have generated these terms? Choose all that apply.
a) #" ^ ^ , and $ "% $ when
.
b) #" ^ &!^ , and $ "%' (^) $ when
.
c) #" ^ (^ ) ^ , and $ "*% +$ %' (^) +$ ) when
!.
d) #" ^ (^ ) ^ , and $ "*%' (^) $ % +$ ) when
!.
Let be a relation on the set ^ ^ ^ . If ^ , then has which of the following properties? (Choose the best answer. Be very careful!)
a) reflexive
b) reflexive and symmetric
c) reflexive, symmetric and transitive
d) reflexive, symmetric, antisymmetric and transitive
The symmetric closure of a relation drawn from a set with cardinality will require (Choose all that apply. Think of the matrix representation of the the relation.)
a) at least elements to be added to the original relation
b) at most ^ $ elements to be added to the original relation
c) as few as zero elements to be added to the original relation
d) at most elements to be added to the original relation
+ # is a poset where ’ ’ means ’divides’. Which of the following statements are true regarding
? (Choose all that apply.)
a) 4 and 6 are incomparable
b)
has a least element
c) 6 and 12 are the maximal elements of
d)
has a greatest element
Short Answer Problems
Express the negations of each of these statements so that all negation symbols immediately precede predicates.
a)
.
(^)
b)
^ ^ 0^
c)
0 )
1
The fibonacci sequence is defined as:
Let be a non-empty set. Define a relation on
, the power set of , as
if and only if
. Is this relation reflexive, symmetric, antisymmetric, transitive and/or a partial order? Justify your answer.
a
d
b
c e
f
The above directed graph represents a relation .
a) List the edges that need to be added to the above graph to form the directed graph representing the transitive closure of .
b) Let
be the transitive closure of the reflexive closure of . Notice
is a poset. Draw its Hasse diagram.
c) What is/are the minimal element(s) of
?
d) What is/are the maximal element(s)?
e) What is the least element (if any)?
f) What is the greatest element (if any)?
g) What are the upper bounds of the set ^ ^?
h) What are the lower bounds of the set ^ ?
Long Problems
The ternary search algorithm searches for an element
in a sorted list as follows. At each step it first compares the element at position ! in the list to
and then possibly compares the element at position ! in the list to
. In every step, it either discovers
and terminates or reduces the size of the list to one-third its current size.
a) Given a single element
and a list of elements that contains
, where is a power of !, devise a recursive algorithm for this search strategy to find the position of
in the list.
b) Analyze the time complexity of your algorithm.
Definition : If is a function with domain , then the relation defined by
iff
is an equivalence relation on , and it is called the kernel relation of .
An interesting consequence of equivalence relations and partitions is that any function can be factored into a com- position of two functions, one an injection, and one a surjection. For a function ) , let
be a partition of by the kernel relation of . Then define the function )^
(^) by ^ ^ and define
) by
^ .
a) Prove that is a surjection.
b) Prove that is an injection.
Draft Paper
