MATH 1800
Fall 2016
Assignment 1 Solutions
Quiz 1, which will take place in class on Friday September 16, will be based on this
material.
Questions 1 to 3 are on material from Wednesday September 7.
1. Determine the cardinality of the following sets:
(a)

MATH 1800
Winter 2017
Quiz 8 Solutions
1. Let A = cfw_1, 2, 3.
(a) (2 marks) Write down a relation R on A that is not reflexive, not symmetric, but it is transitive.
(b) (2 marks) Write down a relation R on A that is reflexive and symmetric,
but not trans

MATH 1800
Winter 2017
Quiz 5 Solutions
1. (5 marks) Let n Z. Prove that if n is odd, then 2n2 + 1 3 (mod 4).
Solution Suppose n is odd. Then n = 2k + 1 for some k Z. We have
2n2 + 1 3 = 2(2k + 1)2 2
= 8k 2 + 8k
= 4(2k 2 + 2k).
Since 4 | (2n2 + 1 3), we ha

MATH 1800
Winter 2017
Assignment 8 Solutions
1. (a) A sequence is defined recursively by a1 = 1, a2 = 2, and an = 3an1 2an2
for n 3. Write out the first few terms of this sequence and guess a formula
for an . Then prove that your guess is correct.
(b) Let

MATH 1800
Winter 2017
Tutorial 10 Solutions
1. (a) Construct the multiplication table for Z8 .
(b) Notice that [0] appears in some of the entries of the table. What is special
about the rows and columns for which [0] is appearing?
(c) What happens if you

MATH 1800
Winter 2017
Tutorial 6 Solutions
1. Disprove the statement: Let n N. If n is prime, then n is odd.
Solution: A counterxample is n = 2 that is prime and is even. Indeeed, this is
the only counterexample.
2. Disprove: Let n Z, n 0. Then 2n2 + 29 i

MATH 1800
Fall 2016
Quiz 8 Solutions
Quiz 8 was written in the tutorial on Friday November 18.
1. (4 marks) Let n N. Use arithmetic in Z11 (that is, use modular arithmetic)
to prove that 11 divides 33n + 10 5n .
Proof. Let n N. Working in Z11 , we have
[3

MATH 1800
Fall 2016
Quiz 5 Solutions
Quiz 5 was written in the tutorial on Friday October 14.
1. (5 marks) Let A, B, and C be sets. Prove that (AB)(AC) = A(B C).
Proof. We first show that (A B) (A C) A (B C). Let x (A
B) (A C). Then x A B and x A C. Thus

MATH 1800
Fall 2016
Quiz 1 Solutions
Quiz 1 was written in class on Friday September 16.
1. Let A = cfw_1, 2.
(a) (1 mark) What is P(A)? (Recall that P(A) is called the power set of A)
(b) (1 mark) Is it true that 1 P(A)?
(c) (1 mark) Is it true that A?
S

MATH 1800
Winter 2017
Assignment 6 Solutions
1. Disprove the statement: For any n Z, n 0, 2n + 3n + n(n 1)(n 2) is
prime.
Solution: We can look for a counterexample using trial and error. Starting at
n = 0, the first few terms given by this formula are 2,

MATH 1800
Winter 2017
Tutorial 7 Solutions
1. Let a, b, c Z with a2 + b2 = c2 . Then at least one of a, b, or c is even.
Proof. Suppose not. Then a, b, and c are all odd, so a = 2r + 1, b = 2s + 1,
and c = 2t + 1 for some r, s, t Z. Then
4r2 + 4r + 1 + 4s

MATH 1800
Winter 2017
Tutorial 11 Solutions
1. Let f : Q Q be defined by g(r) = 5r .
(a) Find g 1 (Z).
(b) Let B = cfw_2k + 1 : k Z. Find g 1 (B).
Solution:
(a) We have
o
n
r
g 1 (Z) = cfw_r Q : g(r) Z = r Q : = n, n Z = cfw_5n : n Z
5
(to get to the last

MATH 1800
Winter 2017
Tutorial 9 Solutions
Problem set
1. Let A = cfw_1, 2, 3 and R = cfw_(1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3, 3).
(a) What is dom(R)?
(b) What is range(R)?
(c) What is R1 ?
(d) Is R reflexive?
(e) Is R symmetric?
(f) Is R transitiv

MATH 1800
Winter 2017
Assignment 7 Solutions
1. Prove that there is no smallest positive rational number.
Proof. Suppose there is a smallest positive rational number q. Since q is rational,
a
we have q = ab for a, b Z, b 6= 0. Note that 2q = 2b
is rationa

MATH 1800
Winter 2017
Quiz 6 Solutions
1. (2 marks) By giving a counterexample, disprove the statement: For any n N,
3 | (2n2 + 1).
Solution: We use trial and error to find a counterexample. When n = 1,
2n2 + 1 = 3, which is divisible by 3. When n = 2, 2n

MATH 1800
Winter 2017
Quiz 9 Solutions
1. (4 marks) Let n N. Use arithmetic in Z7 (that is, use modular arithmetic) to
prove that 7 divides 5 62n + 2.
Proof. We work in Z7 .
n
[5 62n + 2] = [5 62 ] + [2]
= [5 36n ] + [2]
= [5][36n ] + [2]
= [5][36]n + [2]

MATH 1800
Winter 2017
Tutorial 5 Solutions
1. Let a, b Z. If a 5 (mod 6) and b 3 (mod 4), then 4a + 6b 6 (mod 8).
Proof. Suppose a 5 (mod 6) and b 3 (mod 4). Then 6 | (a 5) and
4 | (b 3). Thus a 5 = 6c and b 3 = 4d for some c, d Z. That is, a = 6c + 5
and

MATH 1800
Winter 2017
Assignment 9 Solutions
1. Let A = cfw_a, b, c and B = cfw_w, x, y, z. Let
R = cfw_(a, y), (a, z), (b, x), (b, z)
be a relation from A to B.
(a) What is dom(R)?
(b) What is range(R)?
(c) What is the relation R1 ?
Solution:
(a) dom(R)

MATH 1800
Winter 2017
Tutorial 4 Solutions
1. Let n Z. Then n2 + n + 1 is odd.
Proof. We break in to cases according to whether n is even or odd.
Case 1 : n is even. Then n = 2k for some k Z. We have
n2 + n + 1 = (2k)2 + 2k + 1
= 4k 2 + 2k + 1
= 2(2k 2 +

MATH 1800
Fall 2016
Quiz 10 Solutions
Quiz 10 was written in the tutorial on Friday December 2.
1. For this question, no explanation is required.
How many strings of six of the 26 uppercase English letters (A to Z) are there
(a) (1 mark) if letters may be

MATH 1800
Fall 2016
Quiz 3 Solutions
Quiz 3 was written in the tutorial on Friday September 30.
1. (2 marks) Rewrite the following statement in English.
x R, y R, (x < 0) (y < 0) (xy > 0)
Solution: A rough translation is For all real numbers x and y, if x

MATH 1800
Fall 2016
Assignment 9 Solutions
Quiz 9, which will take place in the tutorial on Friday November 25, will be based on
this material.
Questions 1 to 9 are on material from Wednesday November 16.
1. A function g : Q Q is defined by g(r) = 4r + 1.

MATH 1800
Fall 2016
Assignment 5 Solutions
Quiz 5, which will take place in the tutorial on Friday October 14, will be based on
this material.
Questions 1 to 5 are on material from Wednesday October 6.
1. Let A and B be sets. Prove that A \ B = A if and o

MATH 1800
Fall 2016
Assignment 8 Solutions
Quiz 8, which will take place in the tutorial on Friday November 18, will be based on
this material.
Questions 1 to 5 are on material from Wednesday November 9.
1. Construct the addition and multiplication tables

MATH 1800
Fall 2016
Extra Induction and Prove or Disprove Examples
1. Let n be a natural number n. Then
1 + 3 + 5 + . + (2n 1) = n2 .
2. (Bernoullis inequality) Let n N. Then for all x R,
(1 + x)n 1 + nx.
Recall : The sum of the angles in a triangle is 18

MATH 1800
Fall 2016
Assignment 6b
Assignments are not to be handed in. The solutions will be posted by Friday. Quiz
6, which will take place in the tutorial on Friday November 4, will be based on this
material.
Questions 1 to 4 are on material from Wednes

MATH 1800
Fall 2016
Assignment 2
Assignments are not to be handed in. The solutions will be posted by Tuesday. Quiz
2, which will take place in the tutorial on Friday September 23, will be based on this
material.
Questions 1 to 5 are on material from Wedn

MATH 1800
Fall 2016
Tutorial Problems 1
There is no tutorial this week. The following is a list of problems that would have
been covered in a tutorial, along with full solutions.
1. Give examples of three sets A, B, and C such that
(a) A B C, A 6= B, and

MATH 1800
Fall 2016
Assignment 3 Solutions
Quiz 3, which will take place in the tutorial on Friday September 30, will be based on
this material.
Questions 1 to 4 are on material from Wednesday September 21.
1. Let
P (n) : 2n + 1 is odd
and
Q(n) : n is eve

MATH 1800
Fall 2016
Assignment 10 Solutions
Quiz 10, which will take place in the tutorial on Friday December 2, will be based on
this material.
All questions are on material from Wednesday November 23 and Friday
November 25.
Note: On the quiz and exam, a

MATH 1800
Winter 2017
Tutorial 8 Solutions
1. Note: I recommend defining the recursive sequence below and having the class
try to guess a formula for an (by looking for a pattern among the first few
values), and then prove it is correct. However, students

MATH 1800
Fall 2016
Quiz 4 Solutions
Quiz 4 was written in the tutorial on Friday October 7.
1. (5 marks) Let a, b, c Z with a 6= 0 and b 6= 0. Prove that if a 6 | (a + b + c),
then a6 | b or b6 | c.
Proof. We use a proof by contraposition. Suppose a | b

MATH 1800
Fall 2016
Quiz 2 Solutions
Quiz 2 was written in the tutorial on Friday September 23.
1. Let
P : the product of two negative numbers is positive
Q : the sum of two negative numbers is positive
Determine if each of the following statements are tr

MATH 1800
Fall 2016
Quiz 7 Solutions
Quiz 7 was written in the tutorial on Friday November 11.
1. Let A = cfw_x, y, z.
(a) (2 marks) Write down a relation R on A that is not reflexive, not symmetric, and not transitive.
(b) (2 marks) Write down a relation

MATH 1800
Fall 2016
Quiz 9 Solutions
Quiz 9 was written in the tutorial on Friday November 25.
1. (2 marks) A function f : Z Z is defined by f (n) = 3n3. Determine f 1 (B),
where B = cfw_9k : k Z.
Solution: We have
f 1 (B) = cfw_n Z : f (n) B
= cfw_n Z :