ECE 103 Winter 2013 Final Exam Practice Problems
Practice makes 6, 28, 496, 8128, 33550336, . . .
These problems are intended to help you prepare for the final exam. They are in no way an
indication of the questions and their difficulties on the actual ex
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.
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
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
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
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(
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
ECE 103 Winter 2013 Final Exam Practice Problems
(Abbreviated Solutions)
Practice makes 6, 28, 496, 8128, 33550336, . . .
These are merely abbreviated solutions, intended to point you in the right direction or check your
numerical answers. If you write th
Tutorial Problems 6 (Abbr Solns)
These are abbreviated solutions to the tutorial problems. In assignments and exams, you need to
write full solutions.
1. (a) cfw_(8220000 + 4115n, 3617000 + 18107n) | n Z.
(b) No integer solutions exist.
2. x , y such that
Tutorial Problems 5 (Abbr Solns)
These are abbreviated solutions to the tutorial problems. In assignments and exams, you need to
write full solutions.
1. (a) False.
(b) False.
(c) True.
2. (a) 15. x = 11, y = 2.
(b) 1 and 17.
(c) an = fn+2 , bn = fn+1
3.
Tutorial Problems 7 (Abbr Solns)
These are abbreviated solutions to the tutorial problems. In assignments and exams, you need to
write full solutions.
1. (a) 0
(b) 12
(c) 10
2. 32 9 2 (mod 7)
3. Sum of all the digits is divisible by 9.
4. () Divide by d.
Tutorial Problems 9 (Abbr Solns)
These are abbreviated solutions to the tutorial problems. In assignments and exams, you need to
write full solutions.
1. Isomorphic to H1 , not isomorphic to H2 .
0
0
1
1
7
2
5
2
6
5
4
2. (a)
3
G3
(b) 3k.
(c) () A = cfw_0,
Tutorial Problems 3 (Abbr Solns)
These are abbreviated solutions to the tutorial problems. In assignments and exams, you need to
write full solutions.
1. I, 4
2. 2 possibilities
3. n Z, n 2 . . .
n Z, (n 2) . . .
4. (a) True.
(b) False.
(c) True.
(d) True
Tutorial Problems 8 (Abbr Solns)
These are abbreviated solutions to the tutorial problems. In assignments and exams, you need to
write full solutions.
1. (a) No solutions.
n 29 (mod 36).
(b) gcd(m1 , m2 )|(a2 a1 ),
m1 m2
d .
2. 534
3. Split into 2 congrue
Tutorial Problems 3 (Abbr Solns)
These are abbreviated solutions to the tutorial problems. In assignments and exams, you need to
write full solutions.
1. Pigeons: the distance between a guest and their chair. Holes: possible distances.
2. A person has the
Tutorial Problems 10 (Abbr Solns)
These are abbreviated solutions to the tutorial problems. In assignments and exams, you need to
write full solutions.
1. By contradiction.
2. n odd.
3. (a)
(b) n/2
4. Prims algorithm.
5. By contradiction.
Tutorial Problems 6 (Abbr Solns)
These are abbreviated solutions to the tutorial problems. In assignments and exams, you need to
write full solutions.
1. Check that an element in A (B C) is also in (A B) (A C), and vice versa.
2. (a) False.
(b) False.
(c)
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