Math 299, Section 003
Homework Assignment 4 (due 01/22)
Section 2.4
2.20 (2.15 in 2nd edition) For statement P and Q, construct a truth table for
(P = Q) = ( P ).
2.24 Two sets A and B are nonempty disjoint subsets of a set S . If x S , then which
of the
MTH299 - Homework 5
Name: Ron Durga
Homework 5; Due Thursday, 02/16/2017
Quick Answer Questions. No work needed. No partial credit available.
Question 1. Rewrite the following as if. . . , then . . . statements.
(a) A sufficient condition for x to be posi
Recitation 7 (Solutions)
Math 299
Exercise 1: Prove that there exists a natural number other than 6 that is the sum of its proper
divisors.
Proof. The number 28 is a sum of its proper divisors. Indeed, its proper divisors are
1, 2, 4, 7, 14, and 1 + 2 + 4
Math 299
Homework 6 Solutions
1. (a) Apply A3 with c = a to get a2 ab. Since 0 a and a b,
A1 implies that 0 b. This means that we can apply A3 again
but with c = b to get ab b2 . By A1 we know that a2 ab and
ab b2 implies a2 b2 , as desired.
(b) We have a
Lectures 1 and 2: Chapter 1. Sets
Math 299
Denition:
A set is a welldened collection of distinct objects called the elements (or
members).
Sets are conventionally denoted with capital letters.
1. We enclose a set with braces (curly brackets) cfw_.
2. The
Lectures 8 and 9: Chapter 3
Math 299
0. Ch3.1 A trivial proof and a vacuous proof (Reading assignment)
1. Ch3.2 Direct proofs
2. Ch3.3 Proof by contrapositive
3. Ch3.4 Proof by cases
4. Ch3.5 Proof evaluations (Reading assignment)
Ch 3.2: Direct proofs
A
Lectures 6 and 7: Sec 2.9 and Sec 2.10
Math 299
2.9: Some Fundamental Properties of Logical Equivalence
Theorem: For statements P , Q, and R, the following properties hold.
1. Commutative Laws
(1) P Q Q P
(2) P Q Q P
2. Associate Laws
(1) P (Q R) (P Q) R
Math 299
Lecture 5: Sections 2.6 and 2.7.
Biconditional, Converse, Tautologies and Contradictions
Converse
The statement B A is the converse of the statement A B .
Give an example for each of the following cases:
The statement is true and its converse is
Math 299
Lecture 3: Chapter 2. Logic
Sec 2.1: Statements
Mathematics is the business of proving mathematical statements to
be true or false . Logic lays the foundation for rigorous mathematical
proofs.
Denition: A statement is a sentence that is either tr
Math 299Lecture 4: Chapter 2. Logic. Sections 2.4 and 2.5: Implications
If ., then . Statements
Denition: Statements of the form If statement A is true, then statement B is true.
are called implications. Mathematically this is denoted by A B .
AB
If A th
Lecture 10: Sections 4.1 and 4.2
Math 299
Divisibility of Integers and Congruence of Integers
Denition: Let a, b be non-zero integers. We say b is divisible by a (or a divides b,
or b is a multiple of a) if there is an integer x such that a x = b. And if
Lecture 11: Section 4.3
Math 299
Proofs Involving Real Numbers
Properties:
A1 For all real numbers a, b, c, if a b and b c then a c.
A2 For all real numbers a, b, c, if a b then a + c b + c.
A3 For all real numbers a, b, c, if a b and 0 c then ac bc.
Prov
Math 299
Lecture 19: Section 6.4
Strong Mathematical Induction.
The Strong Principle of Mathematical Induction
For each n N, let P (n) be a statement. If
1. Base step: P (1) is true and
2. Inductive step:
k N, P (1) P (2) P (k ) = P (k + 1) is true,
then
Lectures 17 and 18: Sections 6.1 - 6.2
Math 299
Mathematical Induction.
Denition: A nonempty set S of real numbers is well-ordered if every nonempty
subset of S has a least element.
Example of a well-ordered set:
Example of a set which is not well-order
Lecture 16: Sections 5.4 - 5.5
Math 299
Existence Proofs.
Existence Proofs
Our goal in this section is to prove a statement of the form
There exists x for which P (x). (That is, 9x, P (x).
I. A constructive proof of existence: The proof is to display a sp
Homework 17 Solutions
Math 299
Fall 2013
Problem 0. We will prove by induction that
n
x2 =
x=1
n(n + 1)(2n + 1)
6
for all positive n N. The base case is
1(2)(3)
6
which is obviously true. For the inductive step, assume
12 =
k
x2 =
x=1
k (k + 1)(2k + 1)
6
Induction (Solutions)
Math 299
1. Prove 12 + 22 + . . . + n2 =
n(n+1)(2n+1)
.
6
Proof. We will prove this by induction on n N.
Base Case: When n = 1, the formula reads 12 =
obviously true.
n(2)(3)
,
6
Inductive Step: Let k 1, and suppose 12 + . . . + k 2
MTH299 - Homework 1
Name: Ron Durga
Homework 1; Due Wed, 09/07/2016
Answer-Only Questions. Credit based solely on the answer.
Question 1. In each instance of the set, S, below, compute the cardinality of S (that is, |S|). If the
cardinality of a particula
MTH299 - Homework 5
Name: Ron Durga
Homework 5; Due Wednesday, 10/05/2016
Quick Answer Questions. No work needed. No partial credit available.
Question 1. Rewrite the following as if. . . , then . . . statements.
(a) A sufficient condition for my garden t
MTH299 - Homework 3
Name: Ron Durga
Homework 3; Due Wednesday, 02/03/2016
Answer-Only Questions. Credit based solely on the answer.
Question 1. Construct truth tables for the following:
(a) not(A and B),
Solution.
A B A
T T
T F
F T
F F
and B
T
F
F
F
not(A
MTH299 - Homework 2
Name: Ron Durga
Homework 2; Due Wednesday, 09/14/2016
Answer-Only Questions. Credit based solely on the answer.
Question 1. Describe the shaded regions in the following Venn diagrams using , ,
(a) (A B) \ (A B)
(b) (A B)cfw_ (A B)
(d)
MTH299 - Homework 6
Name: Ron Durga
Homework 6; Due Wednesday, 2/24/2016
Quick Answer Questions. No work needed.
Question 1. Rewrite the following using and .
(a) For all real numbers x, either x < 0 or x 0.
Solution. x R(x < 0 x 0).
(b) There exist two p
MTH299 - Homework 4
Name: Ron Durga
Homework 4; Due Wednesday, 02/10/2016
Answer-Only Questions. Credit based solely on the answer.
Question 1. Suppose that A is true and B is false. Which of the following are true?
(i) A B,
(ii) A A,
(iii) A or not B,
(i
MTH299 - Homework 6
Name: Ron Durga
Homework 6; Due Wednesday, 10/12/2016
Quick Answer Questions. No work needed.
Question 1. Rewrite the following using and .
(a) There exists a real number r such that r = er .
Solution. r R(r = er ).
(b) For all n N, th
MTH299 - Homework 5
Name: Ron Durga
Homework 5; Due Wednesday, 2/17/2016
Quick Answer Questions. No work needed. No partial credit available.
Question 1. Book Exercise 8.13 (i): Rewrite the following as if. . . , then . . . statements.
(a) A sufficient co
MTH299 - Homework 1
Name: Ron Durga
Homework 1; Due Wed, 01/20/2016
Answer-Only Questions. Credit based solely on the answer.
Question 1. State the cardinality of each of the given sets. If the cardinality of a particular set is not
defined, then state wh
MTH299 - Homework 3
Name: Ron Durga
Homework 3; Due Wednesday, 09/21/2016
Answer-Only Questions. Credit based solely on the answer.
Question 1. Construct truth tables for the following:
(a) not(A and (not B),
Solution.
A B A and (not B)
T T
F
T F
T
F T
F
Review Problems for Midterm Exam II MTH 299 Spring 2014
1. Use induction to prove that
1 + 3 + 6 + +
n(n + 1)(n + 2)
n(n + 1)
=
2
6
for all n N.
2. Use induction to prove that 7|(9n 2n ) for every n N.
3. Use the Strong Principle of Mathematical Induction
Math 299
Recitation 7: Existence Proofs and Mathematical Induction
Existence proofs: To prove a statement of the form x S, P (x), we give either a
constructive or a non-contructive proof. In a constructive proof, one proves the statement
by exhibiting a s