MATH215Spring 2011
Homework Set 1 Solutions
Exercises 4-9 pp. 53-54:
4. Prove the following statements concerning positive integers a, b and c.
(i) (a divides b) and (a divides c) a divides (b + c).
(ii) (a divides b) or (a divides c) a divides bc.
Proof
Euler's Formula for Plane Graphs
by LK
Leonhard Paul Euler (1707 - 1783) was a pioneering Swiss
mathematician who spent most of his life in Russia and
Germany. See <http:/en.wikipedia.org/wiki/Leonhard_Euler> for
information about his life and works.
This
Discrete Calculus
by Louis H. Kauffman
0. N o t a t i o n a l W a r n i n g . In these notes xk equals the k-th power of
x, but x(k) = x(x-1)(x-2).(x-k+1). Thus
x(0) = 1
x(1) = x
x(2) = x(x-1)
x(3) = x(x-1)(x-2)
and so on.
1. We are given a function f(n),
Boolean Algebra
Louis H. Kauffman
1 Introduction
The purpose of these notes is to introduce Boolean notation for elementary logic. In
this version of things we use 0 for F (False) and 1 for T (True). Negation is represented
by placing a bar (or overline)
Math 215
Problem Sampler
1. Construct truth tables for the statements (a) (not A) or B , (b) (not A) and B ,
(c) A B .
2. Construct truth tables for (a) not (A or B ), (b) (not A) and (not B ).
Deduce that the two statements are equivalent.
3. Find a stat
Math 215
Problem Sampler
1. Construct truth tables for the statements (a) (not A) or B , (b) (not A) and B ,
(c) A B .
2. Construct truth tables for (a) not (A or B ), (b) (not A) and (not B ).
Deduce that the two statements are equivalent.
3. Find a stat
Project Math 215 - Due Wednesday, April 29,2009.
Instead of a third hour exam, I am asking you to write a paper
(hand-written or typed, as you wish). The paper will have as its title
"Logic, Sets and Mathematics" and you can use any materials or
problems
Take Home Exam - Math 215 - Fall 2011
1. Prove by induction that
1 2 + 22 + 32 + . + n2 = n ( n + 1 ) ( 2 n + 1 ) / 6
for n = 1, 2, 3,. .
2. Let Un be the n-th Fibonacci number as defined by Eccles in
Definition 5.4.2. Show by induction that
U 1 2 + U2 2
Sample Second Exam - Math 215 - Fall 2010
Write all your proofs with care, using full sentences and correct reasoning.
1. Prove
1
2
+
2
22
+
3
23
+ +
n
2n
=2
n+2
2n
for all n = 1, 2, 3, .
Solution. This is a straightforward induction argument. Solution om
Quiz 1 and Answers - Math 215 - Fall 2009
1. Give a complete and careful proof by induction that
20 + 21 + 22 + 23 + . + 2n = 2n+1 1
for n = 0, 1, 2, .
Answer. Let Pn for n = 0, 1, 2, . be the statement
20 + 21 + 22 + 23 + . + 2n = 2n+1 1.
Note that P0 st
Final Exam - Math 215 - Spring 2009 With Selected Solutions
Do problems 1, 2, 3, 4, 5 and choose one more problem from the remaining
seven problems on the exam. Write all your proofs with care, using full
sentences and correct reasoning.
+2
1
1. Prove 2 +
Exam2 - Math 215 - Fall 2010
Write all your proofs with care, using full sentences and correct reasoning.
1. Prove
1
2
+
2
22
+
3
23
+ +
n
2n
=2
n+2
2n
for all n = 1, 2, 3, .
Solution. This is a straightforward induction argument. Solution omitted.
2. (a)
Exam 2 - Math 215 - Fall 2009 With Solutions
Do problems 1, 2, 3, 4 and choose one more problem from problems 5, 6, 7, 8.
Write all your proofs with care, using full sentences and correct reasoning.
1
1. Prove 2 + 222 + 233 + +
Answer. Omitted.
n
2n
=2
n+
Exam1 With Solutions- Math 215 - Fall 2009
1. Let fn denote the n-th Fibonacci number where f0 = f1 = 1 and
fn+1 =
fn + fn1 for n = 1, 2, . Give a complete and careful proof by induction that
2
2
2
2
f0 + f1 + f2 + + fn = fn fn+1
for n = 0, 1, 2, .
Answer
Ma t h 215 - Su p p lement on F init e a nd Infinit e Set s
by LK
We take as given (for this discussion) the natural numbers
N = cfw_1,2,3,4,5,.
and we let Nk = cfw_1,2,3,.,k be the set of the numbers from 1 to k.
It is assumed that the natural numbers ar
Math 215 -Fall 2010 - Assignment #7
Due Wednesday, November 24, 2010.
1. Read Chapters 15,16,17 (we have done all this in class).
Read Chapters 19,20,21 (this will be discussed in the week of
November 15 - 19).
2. page 225. Problems 2,3,5,6
3. For the pri
Series Solutions
Introduction. Throughout these pages I will assume that you are familiar with power
series and the concept of the radius of convergence of a power series.
Let us consider the example of the second order differential equation
; recall that
More Number Theory Proofs
Rosen 1.5, 3.1
Prove or Disprove
If m and n are even integers, then mn is divisible by 4.
The sum of two odd integers is odd.
The sum of two odd integers is even.
If n is a positive integer, then n is even iff 3n2+8 is even.
n2 +
Math 240: Transition to Advanced Math
Deductive reasoning: logic is used to draw conclusions
based on statements accepted as true.
Thus conclusions are proved to be true, provided
assumptions are true.
If results are incorrect, then assumptions need to be
Functions
Functions
Domain and Range
Functions vs. Relations
A "relation" is just a relationship between sets of information.
A function is a well-behaved relation, that is, given a
starting point we know exactly where
to go.
Example
People and their h
Equivalence Relations:
Selected Exercises
Equivalence Relation
Let E be a relation on set A.
E is an equivalence relation if & only if it is:
Reflexive
Symmetric
Transitive.
Examples
a E b when a b ( mod 5 ).
(Over N)
a E b when a is a sibling of
Name: _
Date: _
App Inventor: Lab 2 Simple Calculator
A. Notes
Developers often need to modify both text and numeric data.
Number Systems
Numeric data is represented using the binary (base 2) number system within every computer. Since binary
numbers of an
Principle of
Strong Mathematical Induction
Let P(n) be a statement defined for integers n;
a and b be fixed integers with ab.
Suppose the following statements are true:
1. P(a), P(a+1), , P(b) are all true
(basis step)
2. For any integer k>b,
if P(i) is
2.5 Higher-order Derivative and Implicit Differentiation
Higher-order Derivative:
For a function y f x , the higher order derivatives are:
Second derivative
'
'
'
d dy d 2 y
dy
f ' f ' 2 y ' y '
dx dx dx dx
Third derivative
f ' '
d3y
y ' '
3
dx
Business Finance Investment
Fall 2016
Assignment #3
CAPM, Linear Factor Models, and Equity Returns
Due Date: November 10, 2016
The focus of this assignment is to better understand the CAPM and to implement asset
pricing models using actual data. You will
Discrete Calculus
Brian Hamrick
1
Introduction
n
How many times have you wanted to know a good reason that
n
i=
i=1
n(n + 1)
. Sure, its true by
2
n(n + 1)(2n + 1)
? Well, there
6
i=1
are several ways to arrive at these conclusions, but Discrete Calculus
Assignments. Math 215. Fall 2010.
1. Assignment Number One.
Due Monday, August 30, 2010.
Eccles. Read Chapters 1,2,3.
Read the article on Even/Odd by David Joyce
(available from our website).
Download the article on Peano Axioms from our website
(you do n