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
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)
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
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 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,
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 3.3 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
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
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 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
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 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
Math 136
Assignment 5
1 1
1. Let A =
. Find all 2 2 matrices B M22 (R) such AB = BA.
0 1
1
2
a
2. Let v = b . Let Lv : R3 R3 be the function Lv (x) = v x (cross product). Show
c
that Lv is a linear map, and determine its standard matrix.
3
3. Assume v R
Math 1C03 Introduction to Mathematical Reasoning
Term 2 Winter 20142015
Final exam question on guest lectures
State your student ID number. If your number begins with 13 or 14, answer Version A. If your
number begins with anything else, answer Version B.
Section 5.1 Question 11
Suppose f : A B and S is a relation on B. Rene a relation R on A as follows:
R = cfw_(x, y) A A|(f (x), f (y) S.
(a) Prove that if S is reexive, then so is R.
Answer: Assume that S is reexive. Given any x A we need to prove that
(x
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
Math 1C03 Introduction to Mathematical Reasoning
Term 2 Winter 20142015
Problem Sheet 2: division and euclidean algorithms
to be completed by Monday January 19 2015
1) Assume that a, b, c are integers. Prove the following assertions.
i) If a|b and b|c the
Math 1C03 Introduction to Mathematical Reasoning
Term 2 Winter 20142015
Problem Sheet 1: prime numbers
to be completed by Monday January 12 2015
1)
i) State the denition of a prime number.
ii) State the denition of relatively prime numbers.
iii) Prove or
Math 1C03 Introduction to Mathematical Reasoning
Term 2 Winter 20142015
Problem Sheet 4: proof by induction
to be completed by Friday February 6 2015
n
1) Let P (n) be the assertion
i=1
1
i3 = n2 (n + 1)2 . Prove that P (n) is true for all n 1.
4
n
2)
i i