MATH 13 SAMPLE MIDTERM EXAM SOLUTIONS
Problem 1 (9 points). For the problems on this page, you should show your work, but you
do not need to write proofs.
(a) Give an example of an element of the set
MATH 13 FINAL EXAM SOLUTIONS
WINTER 2014
Problem 1 (15 points). For each statement below, circle T or F according to whether the
statement is true or false. You do NOT need to justify your answers.
T
Math 13 An Introduction to Abstract Mathematics
Neil Donaldson & Alessandra Pantano
December 2, 2015
Contents
1
Introduction
3
2
Logic and the Language of Proofs
2.1 Propositions . . . . . . . . . . .
MATH 13 MIDTERM EXAM
WINTER 2014
Student name:
Student ID number:
Instructions
Books, notes, and electronic devices may NOT be used. These items must be kept
in a closed backpack or otherwise hidden
MATH 13 MIDTERM EXAM
WINTER 2014
Student name:
Student ID number:
Instructions
Books, notes, and electronic devices may NOT be used. These items must be kept
in a closed backpack or otherwise hidden
This document is a compilation of all the topics/definitions/etc suggested by each group. Feel
free to use this list to help you study =] However, please be aware that it is possible that this list
is
3. Suppose that f : A ! B and g : C ! D are bijections. Prove that h : A C ! B D defined
by
h : (a, c) 7!
?
f (a), g(c)
is a bijection.
4. Consider the subset S1 S1 R4: if youre worried about what R4
Recall that R3 = f(x, y, z) : x 2 Rg can be viewed as a Cartesian product R2 R. If A R2, we can
view
A R = f(x, y, z) : (x, y) 2 A, z 2 Rg
as a subset of R3.
1. Let S1 = f(x, y) 2 R2 : x2 + y2 = 1g be
First try question 2.3.7 from the notes. Here is an extension.
Conjecture. Between any two real numbers there exists a rational number.
Let dxe, the ceiling of x, denote the smallest integer greater t
1. For fixed n 2 N and k 2 Z, we have the notation nZ + k = fnz + k : z 2 Zg.
For the rest of this question let A = 3Z+ 1 and B = 2Z.
(a) Find the elements of A \ f0, 1, 2, 3, 4, 5, . . . , 20g in ros
5. Prove that the function
g : [0, 2p) [0, 2p) ! R3 : (q, f) 7!
(2 + cos q) cos f, (2 + cos q) sin f, sin q
is injective.
6. Suppose that f = 0. Show that g(q, 0) describes a circle in the xz-plane. W
e;
Math 13: Introduction to Abstract Mathema 7"
Final Exam (4-4665)
June 6th, 2016
4-6pm
N amt-1:
Student ]d#:
mm! marks = 100 (per question in brackets]
No calculatmzq or uther duct-rum? devices
Un
Math 13
Homework 3
Due in discussion Thu May 11: problem 6 from section 4.4, problems 6,8 from section 5.2 of the text,
and the following:
1. Prove that
n
X
(1)i i2 = (1)n
i=1
n(n + 1)
for all n N.
2
Math 13: Introduction to Abstract Mathematics
Final Exam (44665)
June 6th, 2016
46pm
Name:
Student Id#:
Total marks = 100 (per question in brackets)
No calculators or other electronic devices
Unless o
Odd-Numbered Solutions: Chapter 2
2.1
Propositions
2.1.1 Slight variants in language/grammar are permitted with the following, but the shape should
be as follows.
(a) If you want to grow, then you mus
Math 13
Homework 1
Due in discussion Thu April 13: section 2.1 problems 8, 10, 14, and the following:
1. Is (P Q) (P Q) a tautology? Explain.
2. Suppose that the following statements are true:
Every
Exercises
2.3.1 For each of the following sentences, rewrite the sentence using quantiers. Then write the
negation (using both words and quantiers)
(a) All mathematics exams are hard.
(b) No football
H OMEWORK 8
M ATH 120A (44700) FALL 2016
(No Canvas question this week.)
1. Is the map Z Z ! S3 given by (i, j) 7! (1, 2)i (1, 2, 3)j a homomorphism? Prove your answer.
2. a. Prove that there is no on
H OMEWORK 7
M ATH 120A (44700) FALL 2016
1. Canvas question.
a. How many homomorphisms are there from Z2 ! S3 ? Briefly explain your answer, but you
dont have to give a full proof.
b. How many homomor
SPACE
MATH 13, 5pm Discussion: Week 2 Wed GROUP WORK
Turn in at the end of discussion for participation credit; dont forget to write all member names
on the back of the worksheet!
Each group should tu
MATH 13 DISCUSSION PROBLEMS THURSDAY MARCH
13, 2016
1. Consider the functions f : N N N and g : N N N defined by
f (ha, bi) =
g(ha, bi) =
a
b
Also consider the following sets:
L
D
E
O
= cfw_ha, bi N N
MATH 13 WINTER 2016 HOMEWORK 6
Due: Monday, March 14, 2016
Please turn in at the final exam.
Each group please turn in only one paper. If you prefer to work alone,
that is fine: A group can consist of
Preliminaries
0.1
Laws of Thought
0.1.1 Axiom (Law of Identity). For each object x, x and x are equal.
0.1.2 Axiom (Law of Noncontradiction). Every statement and its negation cannot both be true.
0.1.
Math 13
Homework 2
Due in discussion Thu April 20: problem 10 from section 2.2, problems 6, 8, and 18(a) from section 2.3
of the text (make sure you have the new version - as of Apr 13), and the follo
Lecture 3
October 5, 2017
De Morgans Laws.
Negation of Implication.
Using truth tables, we know that
.
Theorem 2.10:
.
To prove this, we negate both sides:
, which becomes
.
Contradiction and Tautolog
Lecture 4
October 6, 2017
Methods of Proof.
Direct proof: Assume P is true; show Q is true.
Contrapositive: Want to show
; then
Contradiction: Suppose
Assume
is true,
.
is false; then
is true.
is fals
Lecture 8
October 16, 2017
Definition. Let
, not both zero. The greatest common divisor of m and n, written as
, is the largest positive divisor of both
Euclidian Algorithm. To find
,
, then
and
.
and
Lecture 9
October 18, 2017
A set is a collection of objects.
Definition. If an object
If
is not in a,
is in a set
, we write
, say
is an element of
, or a member of
. We say two sets are equal if they