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
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
Math 299
Lectures 12 and 13: Sections 4.4 - 4.6
Proofs Involving Sets
In general, to prove two sets A and B are equal, we need to show that both A B
and B A are true.
A B : x A, x B .
EXAMPLES.
(1H) A B = A B
(2) (A B ) = A B
(3H) Let A = cfw_x : x 1 (mod
Lectures 14 and 15: Sections 5.1 - 5.2
Math 299
Counterexamples. Proof by Contradiction.
Counterexamples
(8x 2 S, P (x) 9x 2 S, P (x)
If the statement, 8x 2 S, P (x), is false, there exists x 2 S satisfying P (x).
Example
8x 2 Z, 9y 2 Z, ( x
1)y = y
Is i
Recitation 4: Quantiers and basic proofs
Math 299
1. Quantiers in sentences are one of the linguistic constructs that are hard for computers
to handle in general. Here is a nice pair of example dialogues:
1. Student A: How was the birthday party after I l
MTH299
Exam 1 Review Session
1. Let n N and I = cfw_1, 2, . . . , n. For i I , dene
Ai = [(i 1)/n, i/n]. Identify the following sets by
writing each as an interval or a union of two
intervals.
7. Let E denote the set of even integers, x Z, and
A(x) be the
MTH299
Exam 1 Review (Solutions)
1. Let n N and I = cfw_1, 2, . . . , n. For i I , dene
Ai = [(i 1)/n, i/n]. Identify the following sets by
writing each as an interval or a union of two
intervals.
(a)
iI
iI
Solution: Suppose a b and b c. Then
there are in
Recitation 4: Quantiers and basic proofs
Math 299
1. Quantiers in sentences are one of the linguistic constructs that are hard for computers
to handle in general. Here is a nice pair of example dialogues:
1. Student A: How was the birthday party after I l
Quiz 4
Math 299
Please answer each question in the space provided. Use complete sentences
and correct mathematical notation to write your answers. You have 20 minutes to complete this quiz.
1. (4 points) Rewrite each statement using English words instead
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
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
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
Review Problems for Midterm Exam II MTH 299 Spring 2014
1. Use induction to prove that
1 + 3 + 6 + +
n(n + 1)
n(n + 1)(n + 2)
=
2
6
for all n N.
Solution: This statement is obviously true for n = 1 since
So assume there is some k 1 for which
1 + 3 + 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
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
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
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
SYLLABUS
TRANSITIONS: MATH 299, SECTION 6
SPRING 2014
Instructors Name: David Duncan
Instructors Email: duncan42[at]math.msu.edu
Instructors Oce: Wells Hall C-315
Oce Hours: TBD
TAs Name: Alex A. Chandler
Lecture Times and Location: M, W 5-6:20, Wells Hal
Math 299, Section 003
Homework Assignment 1 (due 01/13)
Please write your solutions on a dierent piece of paper. Please refer to the course
syllabus for a detailed explanation of how you should write homework solutions and how
they will be graded.
Section
Math 299
Recitation 3
Sept 12, 2013
1. Let the real functions f and g be dened by:
f (x) = x and g (x) = 1 x2 .
a. Find the largest domain for f so that f g is a real function? Write your
answer using proper notation and briey explain why the domain cant
Math 299
Supplement: Axiomatic Systems
Oct 4, 2013
In informal mathematics, we experiment intuititively with specic numbers, shapes,
algorithms, and real-world systems. For each kind of concrete phenomena, we try to
capture their key features in a formal