Math 1C03 Introduction to Mathematical Reasoning
Term 2 Winter 20142015
Problem Sheet 3: congruences and modular arithmetic
to be completed by Wednesday January 28 2015
1) Write down the denition of the statement a b mod m. Then use the denition to decide
Name:
Math 1003 Assignment 1
Complex numbers (Chapter 8)
(b) Find the real and imaginary parts of (2 + 303.
f; Q4393 + '3, 9254161) A: 3Q-23C332 + K5 {5
W4 W b
__g 231 :ZTFC
\\
SLK
:L\G+3{
_ I 9:51
(c) Find the real and imaginary parts
Math 1C03 Introduction to Mathematical Reasoning
Term 2 Winter 20142015
Problem Sheet 7: cardinality
to be completed by Monday March 9 2015
1) The Pigeonhole Principle states that there is no injective function from a set with n + 1 elements
to a set with
Section 4.3 Question 19
Consider the Following putative theorem:
Theorem? Suppose R is a relation on A and dene a relation S on P(A) as
follows:
S = cfw_(X, Y ) P(A) P(A)|x Xy Y (xRy).
If R is transitive, then so is S.
(a) Whats wrong with the following p
Section 5.2 Question 9
Suppose f : A B and g : B C.
(a) Prove that if f is onto and g is not one-to-one, then g f is not one-to-one.
Answer: Suppose that f is onto and g is not one-to-one. We can nd elements
in B to witness the fact that g is not one-to-o
Student Name and Number:
Math 1C03: Introduction to Mathematical Reasoning
Instructor: Deirdre Haskell
McMaster University Final Exam, 17 April 2007
THIS EXAMINATION PAPER CONTAINS TWO PAGES AND FIVE QUESTIONS,
NUMBERED PART I, 13, PART II, 12. YOU ARE RE
Math 1C03 Introduction to Mathematical Reasoning
Term 2 Winter 20142015
Problem Sheet 5: strong induction, rational numbers and real numbers
to be completed by Monday February 23 2015
(enjoy your Reading Week)
1) The Fibonacci sequence is given by taking
Section 2.1 Question 8
Are these statements true or false? The universe of discourse is R.
(a) xy(2x y = 0)
Answer: True.
Given x R, pick y = 2x, then 2x y = 0.
(b) yx(2x y = 0)
Answer: False.
Such a y would have to be equal to 0 = 2 0 and 4 = 2 2 at the
Section 3.2 Question 7
Suppose that x + y = 2y x and x and y are no both zero. Prove that y = 0.
Answer: Suppose for a contradiction that y = 0. Since x and y cannot both be zero
(given), we know that x = 0. However, x + y = 2y x tells us (through rearran
Section 3.3 Question 18
In this problem all variables range over Z, the set of all integers.
(a) Prove that if a|b and a|c then a|(b + c).
Answer: Since a|b there is some integer k such that b = ka. Similarly there is
some integer such that c = a. Therefo
Math 1C03 Introduction to Mathematical Reasoning
Term 2 Winter 20142015
Problem Sheet 6: rationals as congruence classes, properties of functions
to be completed by Monday March 2 2015
1) Recall that we dened the rational numbers formally as the set of co
Section 3.5 Question 12
(a) Prove that for all real numbers a and b, |a| b iff b a b.
Answer: (): Suppose |a| b. We aim to prove that b a and a b.
Clearly a |a| for all a, so a b is true.
Case 1: If a 0, a b is trivial since b 0 as well.
Case 2: If a < 0,
Section 3.1 Question 15
Consider the following theorem.
Theorem. Suppose x is a real number and x = 4. If
2x5
x4
= 3 then x = 7.
(a) Whats wrong with the following proof of the theorem?
Proof. Suppose x = 7. Then
then x = 7.
2x5
x4
=
2(7)5
74
=
9
3
= 3. T
P versus NP
Matt Valeriote
McMaster University
23 January, 2008
Matt Valeriote (McMaster University)
P versus NP
23 January, 2008
1 / 20
Propositional Formulas
Denition
Propositional variables are variables that are allowed to take on the values True
(T)
Math 1C03 Introduction to Mathematical Reasoning
Term 2 Winter 20142015
Problem Sheet 8: complex numbers
to be completed by Monday March 16 2015
1) Review calculations with complex numbers in both cartesian and polar coordinates by doing
problems selected
Section 1.2 Question 8
Use truth tables to determine which of the following formulas are equivalent to each
other:
(a)
(b)
(c)
(d)
(e)
P Q (P Q) (P Q) P Q (P Q) (Q P ) (P Q) (Q P ) P
T T
T
T
T
F
T
T F
F
F
F
F
F
F
T
F
F
T
F T
F F
T
T
T
T
T
Notice that (a)
M ATH 1C03
I NTRODUCTION TO M ATHEMATICAL R EASONING , W INTER 2017
H OMEWORK 4
S AMPLE S OLUTIONS
Required problems (to be handed in):
1. Gilbert-Vanstone, Ch.1, Problem Set (=PS) 1, Problem 74.
If S and T are sets, the statement S = T can be expressed a
M ATH 1C03
I NTRODUCTION TO M ATHEMATICAL R EASONING , W INTER 2017
H OMEWORK 5
DUE : T UESDAY, F EBRUARY 14, 2017
Required problems (to be handed in):
1. Heres a statement that we can prove by the Principle of Mathematical Induction: n! nn for all
positi
M ATH 1C03
I NTRODUCTION TO M ATHEMATICAL R EASONING , W INTER 2017
H OMEWORK 8
DUE : T UESDAY, M ARCH 21, 2017
Let a and b be integers, and m a fixed positive integer. We defined in class that a and b are congruent
modulo m (or that a is congruent to b m
M ATH 1C03
I NTRODUCTION TO M ATHEMATICAL R EASONING , W INTER 2017
H OMEWORK 9
DUE : T UESDAY M ARCH 28, 2017
Required problems (to be handed in):
1. We discussed the set of integers modulo m in class. In this problem you will carefully prove that the se
M ATH 1C03
I NTRODUCTION TO M ATHEMATICAL R EASONING , FALL 2017
H OMEWORK 4
DUE : T UESDAY 31 J ANUARY 2017
Required problems (to be handed in):
1. Gilbert-Vanstone, Ch.1, Problem Set (=PS) 1, Problem 74.
2. Gilbert-Vanstone, Ch. 1, Exercise Set (=ES) 1,
M ATH 1C03
I NTRODUCTION TO M ATHEMATICAL R EASONING , W INTER 2017
H OMEWORK 3
O VERALL COMMENTS FROM TA (S TEVEN )
Basically the only problem that was common among all the assignment questions was a lack of proper
detail and/or explanation of ones thoug
M ATH 1C03
I NTRODUCTION TO M ATHEMATICAL R EASONING , W INTER 2017
H OMEWORK 3
DUE : T UESDAY 24 J ANUARY 2017
Required problems (to be handed in):
1. Gilbert-Vanstone, Ch.1, Exercise Set (= ES) 1, Problem 39.
2. Gilbert-Vanstone, Ch. 1, ES 1, Problem 40
M ATH 1C03
I NTRODUCTION TO M ATHEMATICAL R EASONING , W INTER 2017
H OMEWORK 2
DUE : T UESDAY J ANUARY 17, 2017
Required problems (to be handed in):
1. Suppose a, b are two real numbers. From the fact that a > b, is it possible to conclude that |a| > |b|
M ATH 1C03
I NTRODUCTION TO M ATHEMATICAL R EASONING , W INTER 2017
H OMEWORK 6
DUE : T UESDAY, F EBRUARY 28, 2017
The greatest common divisor of two integers a and b , not both zero, is the largest positive integer
dividing both a and b. For the purposes
As a rough guideline, I'm awarding 1-2 marks for the "right answer", 1 mark for orginazation, neatness, correct English, etc, and the remaining marks for how well you explained and justified your reasoning. "Correct English" doesn't mean that you'll lose
M ATH 1C03
I NTRODUCTION TO M ATHEMATICAL R EASONING , W INTER 2017
H OMEWORK 10
DUE : T UESDAY 4 A PRIL 2017
Required problems (to be handed in):
1. Find complete solutions to the following. Explain your work.
(a) 3x 5 (mod 13).
(b) x2 6x (mod 8). (Hint: