ECE 103 Winter 2013: Assignment 5
Due: 1:20 PM, Tuesday, February 12 2013 in class
Last Name:
First Name:
I.D. Number:
Tutorial section (circle one):
Mark (For the marker only):
101
102
103
/32
1. cfw_6 marks Find the prime power decompositions of 15! and
ECE 103 Winter 2013: Assignment 1
Due: 1:20 PM, Tuesday, January 15 2013 in class
Last Name:
First Name:
I.D. Number:
Tutorial section (circle one):
101
102
Mark (For the marker only):
103
/34
Acknowledgments:
1. cfw_5 marks Let P, Q, R be arbitrary state
ECE 103 Winter 2013: Assignment 8
Due: 1:20 PM, Tuesday, March 12 2013 in class
Note: You may express a probability as either an exact fraction or a percentage rounded to two decimal places.
Last Name:
First Name:
I.D. Number:
Tutorial section (circle one
ECE 103 Winter 2013: Assignment 10
Due: 1:20 PM, Tuesday, April 2 2013 in class
Last Name:
First Name:
I.D. Number:
Tutorial section (circle one):
Mark (For the marker only):
101
102
103
/26
1. cfw_3 marks Draw a 3-regular graph that has a bridge.
2. cfw_
ECE 103 Winter 2013: Assignment 6
Due: 1:20 PM, Tuesday, March 5 2013 in class
Last Name:
First Name:
I.D. Number:
Tutorial section (circle one):
Mark (For the marker only):
101
102
103
/30
1. cfw_6 marks Consider the following steps in solving this simul
ECE 103 Winter 2013: Assignment 9
Due: 1:20 PM, Tuesday, March 29 2013 in class
Last Name:
First Name:
I.D. Number:
Tutorial section (circle one):
Mark (For the marker only):
101
102
103
/30
1. cfw_6 marks The following statements are false. Give a counte
ECE 103 Winter 2013: Assignment 3
Due: 1:20 PM, Tuesday, January 29 2013 in class
Last Name:
First Name:
I.D. Number:
Tutorial section (circle one):
Mark (For the marker only):
101
102
103
/26
1. cfw_5 marks The fibonacci sequence cfw_fn is defined by
f0
ECE 103 Winter 2013: Assignment 4
Due: 1:20 PM, Tuesday, February 5 2013 in class
Last Name:
First Name:
I.D. Number:
Tutorial section (circle one):
Mark (For the marker only):
101
102
103
/29
1. cfw_7 marks Mario has 773500 gold coins to purchase a numbe
ECE 103 Winter 2013: Assignment 11
Due: Never. (Never say never.)
1. Let G be a 3-regular planar bipartite graph with 14 vertices. How many faces does G have? Assuming that
G has faces of degrees 4 or 6, how many faces of degree 4 does G have? Draw one su
ECE 103 Winter 2013: Assignment 2
Due: 1:20 PM, Tuesday, January 22 2013 in class
Last Name:
First Name:
I.D. Number:
Tutorial section (circle one):
101
Mark (For the marker only):
102
103
/42
1. cfw_6 marks Consider the following proposition.
Proposition
Sets and Quantifiers
Wednesday, June 17, 2015
3:54 AM
Sets
A collection of objects, which are the elements of the
set.
We write aA to denote that a is an element of A. We
write a/A to mean NOT(aA).
E.g. 7Z, but 1.55/N.
For a set A, the number of
Prime Numbers
Wednesday, June 17, 2015
1:54 PM
Definition 2.5.1 An integer n 2 is prime if its only positive divisors
are 1 and n itself. If n 2 is not prime, we say it is composite.
Every integer greater than one can be expressed as a product of
p
Strong Induction
Wednesday, June 17, 2015
4:09 AM
Let P(x) be a statement that depends on the variable x, and let n0 and n1 be
given integers with n0 n1. To prove that P(n) is true
(a) (basis step) Prove that the base cases P(n0),P(n0 +1),P(n0 +
2),P(
Statements
Wednesday, June 17, 2015
12:06 AM
Definition 1.2.1 Let P and Q be two statements.
1. The statement P AND Q is called the conjunction of P and Q and is true
when both
P and Q are true and false otherwise.
2. The statement P OR Q is called t
Congruence
Wednesday, June 17, 2015
2:27 PM
Definition 3.1.1 For any integers a, b, and m, m > 0, we say that a is congruent
to b modulo m, and write
ab (modm) if m|(ab).
If m|(ab), we say that a is not congruent to b modulo m and write
a b (mod m).
Basic Proofs
Wednesday, June 17, 2015
1:08 AM
Direct Proof
In Direct proof of x P(x) Q(x) assume P(x) is true and
make a sequence of small deductions (hopefully each of
which is obvious) from that until we can conclude that Q(x)
is also true.
Examp
Divisibility Basics
Wednesday, June 17, 2015
12:35 PM
Definition 2.1.1 We say that the integer a divides the integer b, and write
that a|b, if there is an integer q such that b = qa. If a does not divide b, we
write a|b.
E.g. 4|28 is the statement 4
Euclidean Algorithm
Wednesday, June 17, 2015
1:38 PM
The Euclidean Algorithm: Given nonnegative integers r1 and r2 with r2 > 0, we make
repeated application of the division algorithm to obtain the sequence of equations
r1 = q3r2+r3 where 0 < r3 < r2
r2
Linear Diophantine Equations
Wednesday, June 17, 2015
1:46 PM
Definition 2.4.1 Let a1, a2, . . . , an and m be given integers. If we are only interested
in integer solutions of the linear equation
a1x1 +a2x2 +anxn =m,
where x1, x2, . . . , xn are varia
Basic Formulae
Sunday, July 5, 2015
5:32 PM
1. THE MULTIPLICATION PRINCIPLE
If operation A can be performed in a ways, and for each of these ways
operation B an be performed in b ways, then the combined operation A and
B can be performed in ab ways.
e.
ECE 103 Winter 2013: Assignment 7
Due: 1:20 PM, Tuesday, March 12 2013 in class
Note: You can leave binomial coefficients, large factorials and exponents alone, you do not need to evaluate their
actual values.
Last Name:
First Name:
I.D. Number:
Tutorial
ECE 103 - Assignment 3
Due: Tuesday May 22, 2012 during tutorials (Yes, you have a tutorial after the long weekend!)
1. Prof. Rohs favourite game is the Game of Piles. It starts with a single pile of rocks.
There are a series of moves where a single pile
ECE 103 - Assignment 2
Due: Monday May 14, 2012 during tutorials 1. Let A, B, C be sets in the universe of discourse D. Prove the following pairs of sets are equivalent or not: (a) (A B)\C and (A\C) (B\C) (b) (A\B) C and (A C)\(B C) 2. Assume n N. Prove t
ECE 103 Discrete Math for Engineers
University of Waterloo
Spring 2010
Instructors: Koray Karabina, Ashwin Nayak
Homework 8, due July 19, 2010
The rst four questions pertain to the following class of graph. The odd graph On is the graph whose
vertices are
ECE 103 Discrete Math for Engineers
University of Waterloo
Spring 2010
Instructors: Koray Karabina, Ashwin Nayak
Homework 7, due July 12, 2010
Question 1. [8 marks] Problem 37, page 133 of the text book.
Question 2. [7 marks] Consider families with three
ECE 103 Discrete Math for Engineers
University of Waterloo
Spring 2010
Instructors: Koray Karabina, Ashwin Nayak
Homework 6, due June 28, 2010
Note: All questions carry 5 marks.
Question 1. Exercises 3 and 5, page 130 of the text book.
Question 2. Exercis
ECE 103 Discrete Math for Engineers
University of Waterloo
Spring 2010
Instructors: Koray Karabina, Ashwin Nayak
Homework 5, due June 21, 2010
Note: All questions carry 5 marks.
Question 1. Verify that 61 and 97 are prime. Find all possible values for e w
ECE 103 Discrete Math for Engineers
University of Waterloo
Spring 2010
Instructors: Koray Karabina, Ashwin Nayak
Homework 4, due June 7, 2010
Note: All questions carry 5 marks.
Question 1. Calculate gcd(1137, 419) using the Euclidean algorithm. Show all y
ECE 103 Discrete Math for Engineers
University of Waterloo
Spring 2010
Instructors: Koray Karabina, Ashwin Nayak
Homework 3, due May 31, 2010
Note: The rst three questions carry ve marks, the last carries 10 marks.
Question 1. Let tn denote the number of