Adam Matthews
1) Long-run economic growth and short0run fluctuations in output and
employment that are often referred to as the business cycle. These
phenomena are closely related because they happen simultaneously
2) A slowdown is a decline in the econom
7/28/2010 Section 1.
Alg: Unit 15, Tryout
Tryout Problem 1. Show that the integral polynomials x2 - 3 and x2 + 5 are irreducible over Q.
Proof. If x2 - 3 is reducible over Q, then p(x) = x2 - 3 is divisible by a polynomial of degree 1. So, there is a poly
7/28/2010 Unite 14: Division Algorithm for Polynomials.
Alg: Unit 14
In this Unit we shall prove and study the properties implied by the Division Algorithm for polynomials. So far, we only considered the properties of the polynomial ring F [x] with F an i
9/20/2010
Alg: Unit 5 Supplement
Note: How not to prove an equality.
When showing equality of mathematical expressions, some students discovers an argument which seems logical and intuitive at rst sight, but careful analysis shows it is not.
First, note t
9/12/2010
Alg: Unit 4
Unit 4: Function.
In this Unit, we shall dene the concepts of relation, function and binary operation in terms of the Cartesian
Product of sets. Since these concepts are dened in terms of Cartesian Products, let us recall the denitio
7/28/2010
Alg: Unit 4, tryout
Section 1. Relation.
Tryout Problem 1. Let S = cfw_(r3 , r2 ) : r R.
a. Is 8 S 4? Explain. b. Is 8 S 4? Explain.
c. Is 8 S 4? Explain.
d. Is 8 S 9? Explain.
Solutiion. a. Yes. Note that (8, 4) S , since we can take r = 2.
b.
7/27/2010
Alg: Unit 3
Unit 3. Set Notation.
In this Unit, we review necessary concepts and notations from the set theory for the later Units. We will take
a very naive approach, but enough rigor to follow the discussions in the later Units. For us, a set
7/27/2010
Alg: Unit 3, tryout
Section 1. Membership Condition.
Tryout Problem 1. Let A = cfw_n Q : n = 3.2 m for some m Z. Determine whether the following numbers
belong to the set A or not (with brief reasons).
a. 3.2. b. 1. c. 0. d. 12.8. e. 3/200.
Solu
7/27/2010
Alg: Unit 3, Appendix I
Unit 3, Appendix I: Comments on Proof.
The purpose of a proof is to explain to another person why a certain mathematical statement is true. So,
in the proof, we must form complete English sentences and clearly lay out our
7/28/2010 Unit 15: Irreducible Polynomials and the Degree of Algebraic Numbers.
Alg: Unit 15
In this Unit, we shall study the concept of irreducibility of a polynomial in F [x]. The irreducible polynomials act just like prime numbers in the integers. Also
Due 3/28/2016
CHAPTER 6
An Introduction to Macroeconomics
35. (Last Word) Explain the two popular opinions held by economists on how to improve the economy cycles.
CHAPTER 7
Measuring Domestic Output and National Income
1. Of what use is national income a
Chapter 20 International Trade
US trade deficit in goods
735 billion in 2012
US trade surplus in services
196 billion in 2012
Canada largest US trade partner
Trade deficit with China
315 billion in 2012
Exports are 14% US output
Dependence on oil
Principa
Adam Matthews
Macroeconomics
1. Economics is the social science concerned with how individuals, institutions,
and society make optimal choices under conditions of scarcity and choice.
2. There is no such thing as a free lunch because alternatives must be
1) It enables economists and policymakers to: Assess the health of the economy
by comparing levels of production at regular intervals; Track the long-run
course of the economy to see whether it has grown, been constant, or
declined; Formulate policies tha
1) Economists define and measure economic growth as either: An increase in
real GDP occurring over some time period, or as an increase in real GDP per
capita occurring over some time period.
2) $55,000 $50,000 / $50,000 x 100 = 10%. Real GDP per capita =
1) It enables economists and policymakers to: Assess the health of the economy
by comparing levels of production at regular intervals; Track the long-run
course of the economy to see whether it has grown, been constant, or
declined; Formulate policies tha
1) Business cycles are alternating rises and declines in the level of economic
activity.
2) At a peak, business activity has reached a temporary maximum, here the
economy is near or at full employment and the level of real output it at or
very close to th
Math 122 Review Problems
Allocca
Spring 2016
The following problems are NOT indicative of the exact type of questions on the final exam. Rather,
they are merely for self-assessment. Prior to taking the final exam, all students should be able to
answer the
Adam Matthews
1) Long-run economic growth and short0run fluctuations in output and
employment that are often referred to as the business cycle. These
phenomena are closely related because they happen simultaneously
2) A slowdown is a decline in the econom
Homework 2
Due 2/22/2016
1.What is a brief definition of economics? What are the conditions that give rise to this definition?
2.What do economists mean when they say that there is no free lunch? Give another example to which this
statement applies.
3. Is
8/23/2010
Alg: Unit 2
Unit 2: Mathematical Statements and Open Statements.
In this Unit, we study mathematical statements, and how to negate them. Very unfortunately, the negation of a
mathematical statement does not always follow the intuition of some re
7/27/2010
Alg: Unit 2, tryout
Unit 2: Solutions to Tryout Problems.
Section 2. Quantiers
Tryout
a. For
b. For
c. For
problem 1. Determine the truth value of the following statements.
any odd integer n, 2 divides n + 1.
any integers n and m, the sum n + m
7/27/2010
Alg: Unit 2, Summary
Summary.
The purpose of this section is to learn how to negate a mathematical and open statements. To determine the
truth value of a statement, or to prove the trueness of a statement, we often resort in determining or provi
10/17/2010 Section 1. The Divisibility Relation among Integers. Tryout Problem 1. Let b Z. Show that: If 10 | b, then 10 | bc for any c Z.
Alg: Unit 9, tryout
[Note: Proving the If-then statement "If 10 | b, then 10 | bc for any integer c" means we assume
10/16/2010 Unit 8: Axioms of Integer II.
Alg: Unit 8
As noted previously, we will only use the Axioms of Integers, and statements established to be true using the Axioms of Integers. Also, to prove a statement in homework, we may only use the Axioms of In
10/11/2011 Section 1. Well-Ordering Axiom. Tryout Problem 1. The smallest member of a nonempty subset of N is unique.
Alg: Unit 8 tryout
Solution. To show that the smallest element in S is unique, we first assume that there are two smallest elements in S.
10/14/2010 Unit 7: Axioms of Integers I.
Alg: Unit 7
To prove any mathematical statements, we must start with a set of assumptions which we assume to be true without a proof. We refer to such statements as the Axioms (or, the Postulates in Geometry). Ther
7/27/2010 Section 1. Algebraic Axioms.
Alg: Unit 7, Tryout
Tryout problem 1. Justify each step (between the line a and line b, and so on) in the proof of previous lemma: a. Assume that a + b = a + c. b. -a + (a + b) = -a + (a + c). c. (-a + a) + b = (-a +
7/27/2010 Supplemental Note on Proof.
Alg: Unit 7 Supplemental
Recall that: The purpose of a proof is to convince others that a certain mathematical statement is true. A proof must consist of complete English sentences. At the very beginning of the proof
7/28/2010 Unit 6. Algebraic Structure.
Alg: Unit 6
One of the themes of modern algebra is to compare algebraic structures. For us, an algebraic structure refers to a nonempty set equipped with a binary operation or several (usually two) binary operations.