Math 1120B Winter 2014
Chapter 8
Sections 8.1 and 8.2. Relations and Properties of Relations
We are already familiar with several types of relations in mathematics: x = y,
x < y, x > y, x y mod n, A B, etc. In general, for a pair of sets A and B,
a relati
Math 1120B Winter 2014
Chapter 6
Sections 6.1 and 6.2. Mathematical Induction
Let A be a non-empty set of real numbers. An element m A is called a least
element or minimum if a A, a m.
Example: The subset N R of natural numbers has a least element, namely
Math 1120B Winter 2014
Chapter 5
Section 5.1. Existence and Proof by Contradiction
Some statements of the form x D, R(x) are false, which is equivalent to saying
that some statements of the form x D, R(x) are true. Recalling that such a
statement is logic
Math 1120B Winter 2014
Chapter 2
Section 2.1. Logic
A statement is a declarative sentence which is either true or false (but not both).
Every statement has a truth value, namely true or false. The letters P , Q, and
R are used to denote statements, along
Math 1120B Winter 2014
Chapter 3
Section 3.1. Trivial and Vacuous Proofs
A statement which is taken to be true without proof for the purposes of making
an argument is called an axiom. A statement whose truth has been proven
subject to the truth of some co
Practice Problems For Midterm I
Note: These problems are intended for practice for the exam. Do not assume
that items not covered in these problems will not be tested. The midterm
will cover Chapters 1,2,3,4 (rst 4 assignments) and anything covered in
cla
Review for Midterm II
Note: Warning, this is probably not exhaustive and probably does contain
typos (which Id like to hear about), but represents a review of most of the
material covered in Chapters 4-8 inclusive that we covered in class or on
the assign
Math 1120B Winter 2014
Chapter 9
Sections 9.1 and 9.2. Functions
Suppose A and B are sets. A function f : A B from A to B is a relation
f A B such that every element a A is the rst coordinate of exactly one
pair (a, b) f .
Digression: This denition of fun
Math 1120B Winter 2014
Chapter 4
Section 4.1. Proofs Involving Divisibility of Integers
Given two integers a and b with a = 0, we say that a | b or in words a divides b
whenever there is an integer c such that ac = b. You can think of this as meaning
that
Review for Final Exam
Note: Warning, this is probably not exhaustive and probably does contain
typos (which Id like to hear about), but represents a review of most of the
material covered in Chapters 1-11 inclusive that we covered in class or on
the assig