Discrete Mathematics - Final
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1 Instructions
1. Each question is worth 4 points.
2. Attempt as many questions as you can; you will be given partial credit, as per the poli
Discrete Mathematics - Final (Solutions)
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1 Problems
1. Induction:
Let F (n) denote the nth number in the Fibonacci sequence dened and discussed in class. Show that
n
[F (
Discrete Mathematics - Midterm
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1 Instructions
1. Each question is worth 4 points.
2. Attempt as many problems as you can; you will be given partial credit as per the poli
Discrete Mathematics - Quiz I (Solutions)
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1 Problems
1. Is the following argument valid?
[(P (Q R) (R S ) (S T )] (T P ).
Solution: We rst apply the deduction method to r
Due: Thursday, Oct. 3, at the beginnning of class
Please read the comments on the web page about how to do homeworks before doing this homework.
This is a group homework. In all the problems, you must explicitly state your reasoning to
get credit.
1. (20
Discrete Mathematics - Homework I (Solutions)
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1
Problems
1. Let P and Q be two propositions. Explain the differences between the constructs P Q and P Q.
Solution: The in
Discrete Mathematics - Quiz I (Solutions)
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1
Problems
1. Is the following argument valid:
[P (Q R)] [(P Q) (P R)].
Solution: This is a case in which an intuitive argument
Discrete Mathematics - Homework II (Solutions)
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1 Problems
1. Show using induction that
F (2) + F (4) + . . . + F (2n) = F (2n + 1) 1
Solution: It so happens that the rst
Discrete Mathematics - Homework II (Solutions)
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1 Problems
1. Show using induction that
n
[F (i)]2 = F (n) F (n + 1).
i=1
Solution: BASIS : At n = 1, we have,
=
[F (1)]2
=
Discrete Mathematics - Quiz I
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1 Instructions
1. Each question is worth 3 points.
2. The quiz should be turned in by 4 : 45 pm.
3. Attempt as many questions as you can. Yo
Due: Tuesday, Nov. 12, at the beginnning of class
This is a group homework. Please staple your homework in four pairs. 1 and 2, 3 and 4, 5
and 6, 7 and 8. Hand in each of them to the corresponding stack in class.
1. (10 pts) Let Gn be the graph whose vert
University of Illinois at Urbana-Champaign
Department of Computer Science
Homework Assignment 3
CS 273 Introduction to Theoretical Computer Science
Spring Semester, 2005
Due: Thursday, March 10th, at the beginning of class
Please read the comments on the
Due: Tuesday, Dec. 3, at the beginnning of class
This is a group homework. Please staple your homework in four pairs. 1 and 2, 3 and 4, 5
and 6, 7 and 8. Hand in each of them to the corresponding stack in class.
1. (10 pts) For the languages below, either
CS 273
Homework 3 Solutions
Fall 2002
Homework 3 Solutions
1. Prove by induction that if the annihilator for a recurrence is (E b)n , for any positive integer, n, and
constant, b, the solution to the recurrence is
n
ai = b i
cj ij1 , i N
j=1
in which the
University of Illinois at Urbana-Champaign
Department of Computer Science
Homework Assignment 5
CS 273 Introduction to Theoretical Computer Science
Spring Semester, 2005
Due: April 21st, at the beginning of class
Please read the comments on the web page a
Due: Tuesday, February 22nd, at the beginning of class
Please read the comments on the web page about how to do homeworks
before doing this homework. In all problems, you must explicitly state your
reasoning to get credit. Staple together problems 1, 2, 3
University of Illinois at Urbana-Champaign
Department of Computer Science
Midterm1 Solutions
CS 273 Introduction to Theoretical Computer Science
Fall Semester, 2003
Name:
Netid:
Print your name and netid, neatly in the space provided above; print your nam
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Discrete Mathematics - Scrimmage IV (Solutions)
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1
Problems
1. Show that
n
1
i=1 i(i+1)
=
n
n+1 ,
for all n 1.
Solution: We use induction to solve the problem.
Let P (n) d
Discrete Mathematics - Scrimmage V (Solutions)
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1
Problems
1. Prove or disprove:
A (B C) = (A B) (A C)
Solution:
Set B = and C = A. In this case, the LHS is A ( C) = A = A
Discrete Mathematics - Scrimmage V
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1
Problems
1. Prove or disprove:
A (B C) = (A B) (A C)
2. Let A, B and C denote three sets, where A, B and C are subsets of a set S. Pr
Discrete Mathematics - Homework II
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1
Instructions
1. The homework is due on October 30, in class.
2. Each question is worth 3 points.
3. Attempt as many problems as you c
Discrete Mathematics - Midterm
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1
Instructions
1. Attempt as many problems as you can. You will be given partial credit, as per the policy discussed in class.
2. F (n) den
Discrete Mathematics - Midterm (Solutions)
K. Subramani
LCSEE,
West Virginia University,
Morgantown, WV
cfw_ksmani@csee.wvu.edu
1
Problems
1. Propositional Logic:
(i) Show that the argument (P P ) Q is a tautology, using rules of inference.
(ii) The exclu
solutionSolution
CS 273: Intro. to Theory, Spring 2001
Midterm Solutions
March 6, 2001
1. Basic Counting (20 points total)
A wedding photographer is taking pictures of the bride and groom and
four other couples. He wants to arrange all ten people in a row
1. For each of the following, prove whether or not the set G with the specied operation represents a
group.
(a) G = C, the set of complex numbers, and the operation is standard addition of complex numbers.
(b) G = C, the set of complex numbers, and the op
Due: Wednesday May 4th
Please read the comments on the web page about how to do homeworks before doing this homework. In
all problems, you must explicitly state your reasoning to get credit. Hand in each problem seperately.
(100 pts total)
1. (25 pts) Let
Outline
Functions - Permutation Functions and Counting
K. Subramani1
1 Lane Department of Computer Science and Electrical Engineering
West Virginia University
4 November, 2015
Subramani
Relations and Functions
Outline
Outline
1
Permutation Functions
Subra