mcs-ftl 2010/9/8 0:40 page 189 #195
6
6.1
Directed Graphs
Denitions
So far, we have been working with graphs with undirected edges. A directed edge
is an edge where the endpoints are distinguishedone is the head and one is the
tail. In particular, a direc
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11
11.1
Cardinality Rules
Counting One Thing by Counting Another
How do you count the number of people in a crowded room? You could count
heads, since for each person there is exactly one head. Alternatively, you could
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16
16.1
Independence
Denitions
Suppose that we ip two fair coins simultaneously on opposite sides of a room.
Intuitively, the way one coin lands does not affect the way the other coin lands.
The mathematical concept tha
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14
14.1
Events and Probability Spaces
Lets Make a Deal
In the September 9, 1990 issue of Parade magazine, columnist Marilyn vos Savant
responded to this letter:
Suppose youre on a game show, and youre given the choice o
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9
Sums and Asymptotics
Sums and products arise regularly in the analysis of algorithms, nancial applications, physical problems, and probabilistic systems. For example, we have already
encountered the sum 1 C 2 C 4 C
C
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13
Innite Sets
So you might be wondering how much is there to say about an innite set other
than, well, it has an innite number of elements. Of course, an innite set does
have an innite number of elements, but it turns
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12
Generating Functions
Generating Functions are one of the most surprising and useful inventions in Discrete Math. Roughly speaking, generating functions transform problems about sequences into problems about functions
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10
Recurrences
A recurrence describes a sequence of numbers. Early terms are specied explicitly and later terms are expressed as a function of their predecessors. As a trivial
example, this recurrence describes the sequ
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5
Graph Theory
Informally, a graph is a bunch of dots and lines where the lines connect some pairs
of dots. An example is shown in Figure 5.1. The dots are called nodes (or vertices)
and the lines are called edges.
h
b
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3
Induction
Now that you understand the basics of how to prove that a proposition is true,
it is time to equip you with the most powerful methods we have for establishing
truth: the Well Ordering Principle, the Induction
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7
Relations and Partial Orders
A relation is a mathematical tool for describing associations between elements of
sets. Relations are widely used in computer science, especially in databases and
scheduling applications.
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4
Number Theory
Number theory is the study of the integers. Why anyone would want to study the
integers is not immediately obvious. First of all, whats to know? Theres 0, theres
1, 2, 3, and so on, and, oh yeah, -1, -2, .
mcs-ftl 2010/9/8 0:40 page 5 #11
1
Propositions
Denition. A proposition is a statement that is either true or false.
For example, both of the following statements are propositions. The rst is true
and the second is false.
Proposition 1.0.1. 2 + 3 = 5.
Pro
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2
2.1
Patterns of Proof
The Axiomatic Method
The standard procedure for establishing truth in mathematics was invented by Euclid, a mathematician working in Alexandria, Egypt around 300 BC. His idea was
to begin with ve a
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 27, 2010
Midterm
Problem 1. [10 points] Consider these two propositions: P: ( A B) C Q: (C A) (C B) Which of the following best describes the relationship between P an
6.042/18.062J Mathematics for Computer Science Tom Leighton, Marten van Dijk, and Brooke Cowan
October 21, 2010
Midterm Practice Problems
Problem 1. [10 points] In problem set 1 you showed that the nand operator by itself can be used to write equivalent e
6.042/18.062J Mathematics for Computer Science Tom Leighton, Marten van Dijk, and Brooke Cowan
October 21, 2010
Midterm Practice Problems
Name:
This quiz is closed book, but you may have one 8.5 11" sheet with notes in your own handwriting on both sides.
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 27, 2010
Midterm
Name:
This quiz is closed book, but you may have one 8.5 11" sheet with notes in your own handwriting on both sides. Calculators are not allowed. You
6.042/18.062J Mathematics for Computer Science Tom Leighton and Ronitt Rubinfeld
December 20, 2006
Final
Final Problem 1. [8 points] Prove that for all n N, the following identity holds
n i=1
2
i2 =
n(n + 1)(2n + 1) . 6
Solution. By induction on n 1, with
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
December 17, 2008
Final Exam
Problem 1. [25 points] The Final Breakdown Suppose the 6.042 final consists of: 36 true/false questions worth 1 point each. 1 induction problem wo
6.042/18.062J Mathematics for Computer Science Tom Leighton and Eric Lehman
December 14, 2004
Final Exam
YOUR NAME: :
You may use two 8.511" sheets with notes in your own handwriting on both sides, but no other reference materials. Calculators are not al
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
December 17, 2008
Final Exam
Name:
This final is closed book, but you may have three 8.5" 11" sheets with notes in your own handwriting on both sides. You may not use a calcu
6.042/18.062J Mathematics for Computer Science Tom Leighton and Ronitt Rubinfeld
December 20, 2006
Final
This final is closed book, but you may have three 8.5 11" sheet with notes in your own handwriting on both sides. Calculators are not allowed. You ma
6.042/18.062J Mathematics for Computer Science Tom Leighton and Eric Lehman
December 14, 2004
Final Exam
YOUR NAME:
You may use two 8.511" sheets with notes in your own handwriting on both sides, but no other reference materials. Calculators are not allo
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
December 14, 2010
Final
Name:
This quiz is closed book, but you may have two 8.5" 11" sheets with notes in your own handwriting on both sides. Calculators are not allowed. Yo
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
November 08, 2010
Problem Set 10
Problem 1. [15 points] Suppose Pr cfw_ : S [0, 1] is a probability function on a sample space, S, and let B be an event such that Pr cfw_B > 0
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
November 16, 2010
Problem Set 11
Problem 1. [20 points] You are organizing a neighborhood census and instruct your census takers to knock on doors and note the sex of any chil
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 21, 2010
Problem Set 3
Problem 1. [16 points] Warmup Exercises For the following parts, a correct numerical answer will only earn credit if accompanied by it's deriv
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
November 23, 2010
Problem Set 12
Problem 1. [15 points] In this problem, we will (hopefully) be making tons of money! Use your knowledge of probability and statistics to keep
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 28, 2010
Problem Set 8
Problem 1. [25 points] Find bounds for the following divide-and-conquer recurrences. Assume T (1) = 1 in all cases. Show your work. (a) [5 pts]