This book assumes that you understood precalculus when you took it. So you
used to know how to do things like factoring polynomials, solving high school
geometry problems, using trigonometric identities. However, you probably
This chapter covers special properties of planar graphs.
A planar graph is a graph which can be drawn in the plane without any
edges crossing. Some pictures of a planar graph might have crossing edges,
but its p
Sets of Sets
So far, most of our sets have contained atomic elements (such as numbers or
strings) or tuples (e.g. pairs of numbers). Sets can also contain other sets.
For example, cfw_Z, Q is a set containing two innite sets. cfw_a, b, cfw_c is
This chapter covers innite sets and countability.
The rationals and the reals
Youre familiar with three basic sets of numbers: the integers, the rationals,
and the reals. The integers are obviously discrete, in that theres a b
1. A 1944 study on race relations authored by Gunnar Myrdal and funded by The Carnegie Foundation.
2. At 1500 pages, it details what he saw as obstacles to full participation in American society that AfricanAmericans faced at of the 1940s
Study Guide for Exam 2
This a list of possible IDs for the exam:
Delegate vs. Trustee
Descriptive vs. Substantive Representation
Implicit Racial Message
Race is defined by different people. Sometimes scientists, sometimes Congress,
sometimes other people, sometimes themselves.
Why do we gather racial info at all if it's subjective?
Prop 54 in 2003 -California
Would have amended the Califo
-Edit assignment 1- due wednesday
Ethnic options found by knowledge about ancestors, surname, physical appearance,
and rankings of ethnic groups.
60-75% of multi racial children chose their fathers' ancestry over their mother. 1980
Italians were on
Self interest decision made and not much background reasoning behind decision. / what
have you done for me lately? Past behavior - good indicator of what they will do in the
/best for my family/group/nation/etc
Group interest decision ma
Identities of the Second Generation:
Distinction on how West Indians view themselves in America as they are being
categorized by other people as being Black and not West Indian.
Transition from ethnic identity to racial identity.
September 2, 2011
Discussion Section Notes
Identity & interests
Why is Hispanic the only ethnicity of the census?
More useful than other ethnic information.
Reason for no agreed racial classification system because there is no an effective way
to create i
Discussion about immigration. How many should come in, how many do come in (less
than 2 million), and what to do with these immigrants?
Training and education necessary for immigrants?
How does immigration affect the economy? Would closing or opening the
In this chapter, well see state diagrams, an example of a dierent way to use
State diagrams are a type of directed graph, in which the graph nodes represent states and labels on the graph edges
This chapter covers asymptotic analysis of function growth and big-O notation.
Running times of programs
An important aspect of designing a computer programs is guring out how
well it runs, in a range of likely situations. Designers
This chapter covers how to analyze the running time of algorithms.
The techniques weve developed earlier in this course can be applied to analyze how much time a computer algorithm requires, as a function of the siz
Many mathematical proofs use a small range of standard outlines: direct
proof, examples/counter-examples, and proof by contradiction and contrapositive. These notes explain these basic proof methods, as well as how to
use denitions of new
This chapter covers propositional logic and predicate logic at a basic level.
Some deeper issues will be covered later.
A bit about style
Writing mathematics requires two things. You need to get the logical ow of
ideas correct. And you
So far, weve been assuming only a basic understanding of sets. Its time to
discuss sets systematically, including a useful range of constructions, operations, notation, and special cases. Well also see how to compute the sizes of
sets and p
Weve now covered most of the basic techniques for writing proofs. So were
going to start applying them to specic topics in mathematics, starting with
Number theory is a branch of mathematics concerned with the behavi
Mathematical relations are an extremely general framework for specifying
relationships between pairs of objects. This chapter surveys the types of
relations that can be constructed on a single set A and the properties used
Functions and onto
This chapter covers functions, including function composition and what it
means for a function to be onto. In the process, well see what happens when
two dissimilar quantiers are nested.
Were all familiar with fu
Functions and one-to-one
In this chapter, well see what it means for a function to be one-to-one and
bijective. This general topic includes counting permutations and comparing
sizes of nite sets (e.g. the pigeonhole principle). Well also see the
Graphs are a very general class of object, used to formalize a wide variety of
practical problems in computer science. In this chapter, well see the basics
of (nite) undirected graphs, including graph isomorphism, connectivity, and
This chapter covers mathematical induction.
Introduction to induction
At the start of the term, we saw the following formula for computing the
sum of the rst n integers:
Claim 38 For any positive integer n, n=1 i =
This chapter covers recursive denition, including nding closed forms.
Thus far, we have dened objects of variable length using semi-formal denitions involving . . . For example, we dened the summation
This chapter covers trees and induction on trees.
Trees are the central structure for storing and organizing data in computer
science. Examples of trees include
Trees which show the organization of real-world data: family
Psychchological boundaries -Rethinking Assimilation- where do we decide who is
American and who is someone else?
Physical boundaries between countries. Ex. Mexico to US
Segmented assimilation- First and second generation immigrants acce
Monday September 12, 2011
-Born abroad children automatically citizens?
Require English to be a citizen?
Require test of American history, government and civics?
Require living in the US for more than 5 years?
Is it rational to vote?
If PB + D > C , then vote
PB = probability that you will be influential
D = duty, values
C = cost
Vote Turnout dramatically differs by country
Australia is the highest (95%)
European countries roughly speaki