Math 280
Strategies of Proof
Homework Solutions
3.4 Proving Conditional Statements I: Direct Proofs
1. Let f : ( 1, 1] ! R be the function dened by
1)2
f (x) = (x
for each x 2 ( 1, 1].
(a) Show that f is injective.
In symbolic form, we need to show
8x1 2
Math 280
Strategies of Proof
Homework Solutions
3.2 Proving Existentially Quantied Statements
Fall 2012
1. An integer n is composite provided that there exist positive integers r and s such that n = rs and
r 6= 1 and s 6= 1. Write out the two parts of a p
Math 280
Strategies of Proof
Homework Solutions
3.2 Proving Existentially Quantied Statements
Spring 2014
2. Disprove: For all integers n, 5n + 1 is odd.
This has symbolic form
(8n 2 Z) [5n + 1 is odd].
To disprove this, we need to nd a counterexample. Ch
Math 280
Homework Solutions
2.3 Sets
Strategies of Proof
Spring 2014
1. The following sets are written using the rule method. For each set, identify the domain and rule and
write the set using the list method.
(a) A = cfw_n 2 Z | n is a product of two dis
Math 280
Strategies of Proof
Spring 2016
Homework Assignment
Directions for Writing Proofs
You must follow the directions for writing proofs contained in section 3.1 of the class notes (see pages
6667). In particular, here are some guidelines you should b
Math 280 Strategies of Proof Class Notes 2.1 Propositional Functions and Quantied Statements
Fall 2007
In Chapter 1, we dened a statement, or a proposition, as a declarative sentence which has a denite truth value. Most sentences which we encounter in mat
Math 280
Strategies of Proof
Homework Solutions
3.4 Proving Conditional Statements I: Direct Proofs
Fall 2015
1. Let f : ( 1, 0) ! R be the function defined by f (x) = 2
x2 for each x 2 ( 1, 0).
(a) Show that f is injective.
In symbolic form, we need to s
Math 280 Class Notes 3.10 Epsilon-Delta Proofs
Strategies of Proof
Fall 2007
The denitions of the limit of an innite sequence and of the limit of a function are among the most important in mathematics. They are also one of the greatest sources of diculty
Math 280
Strategies of Proof
Homework Solutions
2.1 Predicates and Quantied Statements
Fall 2013
3. Given the following propositional functions with domain Z, the set of all integers
p(n) : n is positive.
q(n) : n is even.
r(n) : n is a perfect square.
s(
Math 280
Homework Solutions
3.5 Proofs in Set Theory
Strategies of Proof
Spring 2014
3. Disprove: For all sets A, B, and C, if A [ C B [ C, then A B.
This statement has symbolic form
8A 8B 8C [(A [ C B [ C) ! (A B)].
To disprove this statement, we need to
Math 280
Strategies of Proof
Spring 2016
Exam 2 Information
Date: Wednesday, March 23
Exam 2 will cover the following sections:
Chapter 2, Sections 2, 3
Chapter 3, Sections 1, 2, 3
You will need to bring a large (8 12 11) blue book to the exam.
For all pr
Math 280
Strategies of Proof
Homework Solutions
3.9 Proof by Mathematical Induction
Fall 2013
12. The Fibonacci sequence f1 , f2 , f3 , . . . is dened recursively by
f1 = 1, f2 = 1,
fn = fn 1 + fn 2
for each integer n
3
Prove: For all integers n 2 N,
+ +
Math 280
Strategies of Proof
Spring 2016
Exam 1 Information
Date: Wednesday, February 24
Exam 1 will cover the following sections:
Chapter 1, Sections 1, 2, 3
Chapter 2, Section 1
You will need to bring a large (8 12 11) blue book to the exam.
Exam 1 Revi
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Math280
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16
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you
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futl credit.
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fullcredit. EXA
Math 280
Course Information
Instructor: Dr. Ernie Solheid
Office: McCarthy Hall 182A
Email: [email protected]
Office Hours: MW 2:30 3:30
WF 10:0010:50
Strategies of Proof
Fall 2016
Phone: (657) 278-7023
Available at other times
by appointment
Course
Math 280
Syllabus
Strategies of Proof
Fall 2016
You should view the indicated video lectures before the dates for which they are listed.
Assignment
Date
Section Video Lecture
due
Aug 22 M
1.1
Statements and Logical ConnectivesParts 1 & 2
24 W
1.1
Statemen
Math 280
Strategies of Proof
Homework Solutions
3.1 Proving Universally Quantified Statements
Spring 2017
2. The Goldbach Conjecture states that every even integer n
numbers. This has symbolic form
(8n) p(n),
4 can be written as the sum of two prime
where
Math 280
Strategies of Proof
Fall 2016
Class Notes
3.10 Epsilon-Delta Proofs
The definitions of the limit of an infinite sequence and of the limit of a function are among the most
important in mathematics. They are also one of the greatest sources of diff
Math 280
Strategies of Proof
Spring 2017
Homework Assignment
Academic Integrity
Students who violate university standards of academic integrity are subject to disciplinary sanctions,
including failure in the course and suspension from the university. Acad