Download Iinductive Definition - Discrete Mathematics - Exam and more Exams Discrete Mathematics in PDF only on Docsity!
CS 173: Midterm Exam 2
Fall 2004
Name:
NetID:
Lecture Section:
General Directions
- Make sure your name is on every page.
- There are 10 pages, including a sheet of scratch paper. Make sure that you answer all 13 questions.
- Remember to write clearly and legibly. Unreadable answers will receive no credit.
- This is a closed book exam. No notes of any kind are allowed.
- Remember to time yourself.
Question Points Out of
1 5
2 5
3 5
4 5
5 5
6 5
7 5
8 5
9 10
10 10
11 10
12 10
13 20
Total 100
Multiple Choice
Problem 1 (5pts)
Given the inductive definition:
,
, and
for
,
is:
a) 8
b) 14
c) 34
d) 36
Problem 2 (5pts)
is a collection of strings of symbols. It is recursively defined by
1.! and " belong to.
- if string # belongs to , so does #$".
Which of the following does NOT belong to?
a) !%"&"'"
b) "&"&"
c) "(!
d)!
Problem 7 (5pts)
When sorted in decreasing order of growth rate, which one of the following functions would be first?
a)
VN,&?L/90 2 W6
b) X
RYZ-/102
c) [
Q
Z?S^]_
[
QZ?S W_
[
Q@S`G_
Q
[
Q\S -_
[
Q\SaG_
d)
H
Z - Vb/102c - ,
Problem 8 (5pts)
How many numbers must be selected from the set d
IF
X
FGYIFGeIF'f5IF6f %F6 X
F'6YIF'6eIFG Z5%g in order to guarantee that at least
one pair adds up to 22?
a) 5
b) 6
c) 7
d) 8
Short Answer Problems
Problem 9 (10pts)
We say that a circle is a positive integer circle if it is centered at
5IF@
and has radius h, where
and h are positive
integers. Show that the set of positive integer circles is countably infinite.
a) Using an indirect proof, show that the set of positive integer circles is infinite.
b) Prove that the set of positive integer circles is countable by defining a 1 to 1 function from the set of circles to
the set of natural numbers.
Problem 11 (10pts)
Prove using induction that every convex polygon with
tuH vertices has exactly
Q
[
Q
sr W_
(A convex polygon is a polygon with the property that every line segment drawn between any two points inside the
polygon lies entirely inside the polygon.)
Problem 12 (10pts)
Prove that in any set of 11 integers, there are two whose difference is divisible by 10.
SCRATCH PAPER
