Rev. Confirming Pages
CONTENTS
MAP 1.2:The Ancient Near East, ca. 1450 B.C.E.
21
THINKING ABOUT ART: Egyptian Fresco,
ca. 12951186 B.C.E. 22
Political Expansion: The New Kingdom,
15701085 B.C.E. 22
The Religious Experiment of Akhenaten,
ca. 13771360 B.C.E
A HISTORY OF WESTERN SOCIETY, 11E
CORRELATION GUIDE
SECTION II: ALIGNING AP
EUROPEAN HISTORY
THEMES WITH A HISTORY
OF WESTERN SOCIETY
The AP European History Course Description
provides a list of important themes that should be
covered in an AP European H
COURSE OUTLINE
HIS 102
Course Number
History of Western Civilization Since 1648
Course Title
3
_
Credits
3
_
Hours: lecture/laboratory/other (specify)
Catalog description:
An introduction to the political, social, cultural, and economic events that have d
NOVA COLLEGE-WIDE COURSE CONTENT SUMMARY
HIS 102 HISTORY OF WESTERN CIVILIZATION II (3 CR.)
Revised 5/2013
Course Description
Examines the development of western civilization from ancient times to the present. Part II of II. Lecture 3
hours per week.
Gene
To Heynes definition: Justice is Fulfillment of legitimate
expectations.
Justice Example: Studying in school, graduating from high school,
etc.
Un-justice Act Example: Not washing ones hands after using the
bathroom.
To me: Justice is fairness, equitabl
What has caused the decline of the middle class?
- Due to the article, there are tax problem, income distribution.
The remarkable similarity in income distribution across countries
over the past century means domestic policy has less effect than
many beli
Introduction to Discrete math
Unit 3 exam review
1. A lottery ticket consists of 4 distinct numbers chosen from 150, and a power ball chosen from a separate set of the numbers
1-60. How many choices of numbers for a ticket are possible?
2. Ping pong balls
Math 201 Khoi Nguyen
Section 7.1
Problems 5
What is the probability that the sum of the numbers on two dice is
even when they are rolled?
Each dice has 6 outcomes.
Total outcomes: 6.6 = 36 ways
Sum of the numbers on two dice is even:
Case 1: both are even
Math 201 Khoi Nguyen
1.
Use a truth table to show that P V (Q R) R V (Q P)
P Q R QR
P V (Q R)
T T T T
T
T T F F
T
T F T T
T
T F F T
T
F T T T
T
F T F F
F
F F T T
T
F F F T
T
=> P V (Q R) R V (Q P)
QP
T
T
T
T
F
F
T
T
R V (Q P)
T
T
T
T
T
F
T
T
2.
Prove the
Math 201 Khoi Nguyen
Section 6.1
Problems 1
There are 18 mathematics majors and 325 computer science majors
at a college.
a) In how many ways can two representatives be picked so that
one is a mathematics major and the other is a computer science
major?
U
Math 201 Khoi Nguyen
Section 6.3
Problems 17
How many subsets with more than two elements does a set with 100
elements have?
Each element has two states, exist or not exist. Thus, in total, there
are 2100 subsets.
By subtracting from the total number of s
Math 201 Khoi Nguyen
1.
A lottery ticket consists of 4 distinct numbers chosen from 1- 50,
and a power ball chosen from a separate set of the numbers 1-60.
How many choices of numbers for a ticket are possible?
4 distinct number from 1-50: C (50, 4) ways
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
R
T
F
T
F
T
F
T
F
QR
T
F
T
T
T
F
T
T
P V (Q R)
T
T
T
T
T
F
T
T
QP
T
T
T
T
F
F
T
T
R V (Q P)
T
T
T
T
T
F
T
T
^ l and, V l or, l xor l ch mt ci ng , l not ; l if and only if
A is necessary for B l B => A ; A is sufficient
Math 201 Khoi Nguyen
1.
State the Division Algorithm and use it to find q and r for each of
the following pairs of a and b
Given integers a and b, b 0, there exists unique integers q and r
such that
a = b. q + r and 0 r < |b|
(a)
a = 200; b = 7
Quotient(q
Math 201 Khoi Nguyen
Section 6.4
Problems 3
Find the expansion of (x + y)6.
6
(6k ) x6 k y k = x6 + 6.x5. y + 15.x4.y2 + 20.x3.y3 + 15.x2.y4 + 6.x.y5 +
k=0
y6
Problem 12
The row of Pascals triangle containing the binomial coefficients
(10k )
, 0 k 10, is
Math 201 Khoi Nguyen
Section 6.2
Problems 4
A bowl contains 10 red balls and 10 blue balls. A woman selects
balls at random without looking at them.
a) How many balls must she select to be sure of having at least
three balls of the same color?
The minimum
Math 201 Khoi Nguyen
Section 4.4
Problems 20
Use the construction in the proof of the Chinese remainder theorem
to find all solutions to the system of congruences x 2 (mod 3), x
1 (mod 4), and x 3 (mod 5).
Ai
2
1
3
mi
Mi = M/mi Inverse yi of Mi modulo mi
Math 201 Khoi Nguyen
Section 7.2
Problem 3
Find the probability of each outcome when a biased die is rolled, if
rolling a 2 or rolling a 4 is three times as likely as rolling each of the
other four numbers on the die and it is equally likely to roll a 2 o
Math 201 Khoi Nguyen
Page 273
Problem 29 (use the algorithm this time)
Find gcd(92928, 123552) and lcm(92928, 123552), and verify that
gcd(92928, 123552) lcm(92928, 123552) = 92928 123552.
[Hint: First find the prime factorizations of 92928 and 123552.]
1
Math 201 Khoi Nguyen
Exam 2 practice
1.
Show how you would organize a direct proof to prove the claim "If
the moon is made of cheese, then the sky is red"
Proof:
Supposed that the moon is made of cheese.
.
.
The sky is red.
2.
Show how you would organize
Math 201 Khoi Nguyen
Page 284
Problem 6
Find an inverse of a modulo m for each of these pairs of relatively
prime integers using the method followed in Example 2.
a a = 2, m = 17
17 = 2. 8 + 1
2 = 1. 2 + 0
1 = 17. 1 + 2. (-8)
-8 is an inverse of 2 modulo
Math 201 Khoi Nguyen
Section 6.5
Problems 1
In how many different ways can five elements be selected in order
from a set with three elements when repetition is allowed?
Because repetition is allowed and selected elements are in order,
each of five element
Introduction to Discrete
Exam 2 practice
1. Show how you would organize a direct proof to prove the claim
If the moon is made of cheese, then the sky is red
2. Show how you would organize an indirect proof to prove the
claim If 2+2=5, then triangles have
Math 201 Khoi Nguyen
Page 244
Problems 5
Show that if a | b and b | a, where a and b are integers, then a = b or
a = b
a | b and b | a imply that there are integers x and y such that a = x. b
and b = y. a. Therefore, a = x. y. a.
Because a 0, we conclude
Discrete Mathematics
1-1. Logic
Discrete Mathematics. Spring 2009
1
Foundations of Logic
Mathematical Logic is a tool for working with
complicated compound statements. It includes:
A language for expressing them.
A concise notation for writing them.
A met
Set Theory
CSE 215, Foundations of Computer Science
Stony Brook University
http:/www.cs.stonybrook.edu/~cse215
Set theory
Abstract set theory is one of the foundations of mathematical
thought
Most mathematical objects (e.g. numbers) can be defined in
te