Carleton University
COMP1805, Fall 2013
Assignment 2 (Solutions)
Due Wednesday, October 16 at 10:00 am
1. Determine whether or not the following arguments are valid. If they are valid, then
state the rules of inference used to prove validity. If they are
COMP1805, Fall 2015
Assignment 2
Due Friday, October 9 at Noon
1. Determine whether or not the following arguments are valid. If they are valid, then
state the rules of inference used to prove validity. If they are invalid, outline precisely
why they are
COMP1805 Tutorial 2
Fall 2015
For more practice problems (with solutions) can be found on the exercise database
http:/cg.scs.carleton.ca/discmath/
1. Prove that the proposition (r p) (q p) (r q) (p q) is a
tautology using the basic equivalence relations (
COMP1805, Fall 2015
Assignment 2
Due Friday, October 9 at Noon
Submit your solution in the dropbox for COMP1805 located in HP3115.
Only submit part (b) of each question.
Your grade will be determined by submitting a reasonable eort for each of the probl
Carleton University
COMP1805, Fall 2013
Assignment 2 (Solutions)
Due Wednesday, October 16 at 10:00 am
1. Determine whether or not the following arguments are valid. If they are valid, then
state the rules of inference used to prove validity. If they are
Question 1: Prove or disprove each of the following propositions.
a) x + y = x + y, for all real numbers x and y.
b) x = x, for all real numbers x.Answer
(Hint: If x R, then either x Z or x = a + r, where a Z and 0 < r < 1.)
Marking scheme
5 marks in tota
mcs 2013/1/10 0:28 page i #1
Mathematics for Computer Science
revised Thursday 10th January, 2013, 00:28
Eric Lehman
Google Inc.
F Thomson Leighton
Department of Mathematics
and the Computer Science and AI Laboratory,
Massachussetts Institute of Technolog
COMP 1805 Discrete Structures
Assignment 3 Solutions
1. Compute the exact sums of the following:
n
n16
1=
(a)
i=17
1
i=1
= n 16
89
89
19 = 19
(b)
j=22
1
j=22
68
1
= 19
j=1
= 19 68
= 1292
k2
(k 2 n + 3) =
k=1
n
2
=
k n
k=1
3
n+
k=1
k=1
n
k=1
n
n
n
n
(c)
COMP 1805 Discrete Structures I
Tutorial 3
January 24, 2012
1. Determine which of the following arguments are valid and which are invalid. Explain why.
(a) Someone in this class has been to Toronto. Everyone who visits Toronto goes up the CN Tower.
Theref
COMP 1805 Discrete Structures I
Assignment 1 (Revised version)
Date Due: Monday, January 28, 2013
Time Due: Before 3pm in the Boxes in Herzberg Room 3115
Write down your name and student number on every page.
Write the course number and section on the r
COMP 1805 Discrete Structures I
Assignment 3
Date Due: March 7, 2013
Time Due: Before 14:00 in the Boxes in Herzberg Room 3115
Write down your name and student number on every page. The questions should be answered in order and your
assignment sheets must
COMP 1805 Discrete Structures I
Tutorial 1
January 9, 2012
1. Are the following sentences propositions? Why or why not?
(a) Always attend the tutorials.
(b) Is today Tuesday?
(c) Today is Thursday.
(d) 2 + 3 = 7
(e) x + 3 = 7
2. Which of the following imp
COMP 1805 Discrete Structures I
Assignment 2
Date Due: Feb 15, 2013
Time Due: Before 2pm in the Boxes in Herzberg Room 3115
Write down your name and student number on every page. The questions should be answered in order and your
assignment sheets must be
COMP 1805 Discrete Structures I
Assignment 5
Date Due: April 10, 2013
Time Due: Before 2pm in the Boxes in Herzberg Room 3115
Write down your name and student number on every page. The questions should be answered in order and
your assignment sheets must
COMP 1805 Discrete Structures I
Assignment 4
Date Due: March 26, 2013
Time Due: Before 2pm in the Boxes in Herzberg Room 3115
Write down your name and student number on every page. The questions should be answered in order and your
assignment sheets must
COMP 1805 Discrete Structures I
Assignment 5
Date Due: April 10, 2013
Time Due: Before 2pm in the Boxes in Herzberg Room 3115
Write down your name and student number on every page. The questions should be answered in order and
your assignment sheets must
COMP 1805 Discrete Structures I
Assignment 4
Date Due: March 26, 2013
Time Due: Before 2pm in the Boxes in Herzberg Room 3115
Write down your name and student number on every page. The questions should be answered in order and your
assignment sheets must
COMP/MATH 1805 Discrete Structures
Assignment 3
Due June 24 at the beginning of class
Write down your name and student number on every page. The questions must be answered in order and
your assignment sheets must be stapled. All questions (or subquestions
COMP/MATH 1805 Discrete Structures Tutorial 1
May 12, 2009
1. State the converse, inverse and contrapositive of the implication "I will use an iPod only if it has a 100 GB hard drive" both logically and in English. Solution: Let a be the proposition "I wi
COMP/MATH 1805 Discrete Structures Tutorial 2
May 19, 2009
1. Recall that, for nite universes of discourse, the quantier notations and are essentially shorthand for conjunctions and disjunctions. If the universe of discourse consists only of the three peo
COMP/MATH 1805 Discrete Structures
Tutorial 5
June 23, 2009
1. Give an example of a relation over cfw_a, b, c, d that is:
(a) reexive and symmetric
Solution: cfw_(a, a), (b, b), (c, c), (d, d), (a, c), (c, a)
(b) symmetric but not reexive
Solution: cfw_(a
COMP/MATH 1805 Discrete Structures
Tutorial 4
June 16, 2009
1. Determine if f (n) is O(g(n) or f (n) is (g(n) or f (n) is (g(n).
(a) f (n) = n2 3n + 4, g(n) = n2
Solution: First note that f (n) is (g(n) since the leading exponents in both polynomials matc
COMP/MATH 1805 Discrete Structures
Tutorial 3
June 2, 2009
1. Let x1 , x2 , . . . , xk be k integers. Let A be the average of these integers. Prove by contradiction that
there must be at least one integer xi whose value is at least A.
Solution: Proof by c
SET: THEORY, OPERATIONS,
AND IDENTITIES
Discrete Structures I
COMP1805
Todays Class
2
Set Theory
What
is a set?
Set Membership
Set Inclusion
Venn Diagrams
Set Operations
Cartesian Product
Operations
Identities
Set Theory Fundamentals
3
Set: Unordered col
FORMAL PROOF METHODS
CONTINUED
Discrete Structures I
COMP1805
Recap: Proof Summary from Last Class
2
Direct proof: p q
Assume
the antecedent, p, is true
Use p and other facts to show that q is true
Indirect proof: use contrapositive q p
Assume
the ante
GRAPHS AND GRAPH
REPRESENTATION
Discrete Structures I
COMP1805
Updates
2
Quiz 2 is over
Assignment 2 due on Tuesday
Quiz 3 next Friday
Last
PollEV
quiz!
Last Class
3
Set Theory
What
is a set?
Set Membership
Set Inclusion
Venn Diagrams
Set Operations
Ca
PLANARITY AND
GRAPH COLOURING
Discrete Structures I
COMP1805
Updates
2
Assignment 2 out of the way!
Assignment 3 due in next Thursday
Quiz 3 on Friday
Last
Quiz!
Final exam has not yet been scheduled
Last Week: Handshaking Theorem
3
For a Graph G = (V,