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 t
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
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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 injectiv
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 transit
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 e
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 Read
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
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
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. S
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
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
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
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 var
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 car
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
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, P
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
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 Pr
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
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 clas
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, Proble