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CS 70
Discrete Mathematics for CS
Fall 2006
Lecture 1
Course Outline
CS70 is a course about on ”Discrete Mathematics for Computer Scientists”. The purpose of the course is to
teach you about:
•
Fundamental ideas in computer science:
 Boolean logic.
 Uncomputability and the halting problem.
 Modular arithmetic, Errorcorrecting codes, secret sharing protocols
 Graphs: paths, cuts, hypercubes.
Many of these concepts underly all the more advanced courses in computer science.
•
Precise, reliable, powerful thinking:
 Proofs of correctness. These are essential to analyzing algorithms and programs.
 Induction and recursion.
 Probability theory.
•
Problem solving skills:
 These are emphasized in the discussion sections and homeworks.
Course outline (abbreviated).
• Propositions, Propositional logic and Proofs
• Mathematical Induction, recursion
• The stable marriage problem
• Modular arithmetic, the RSA cryptosystem
• Polynomials over finite fields and their Applications: errorcorrecting codes, secret sharing
• Graphs: Eulerian paths, hypercubes.
• Diagonalization, SelfReference, and Uncomputability
• Probability and Probabilistic Algorithms: load balancing, hashing, expectation, variance, Chebyshev
and Chernoff bounds, conditional probability, Bayesian inference, law of large numbers, power laws.
CS 70, Fall 2006, Lecture 1
1
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View Full DocumentLesson Plan
In order to be fluent in mathematical statements, you need to understand the basic framework of the language
of mathematics. This first week, we will start by learning about what logical forms mathematical theorems
may take, and how to manipulate those forms to make them easier to prove. In the next few lectures, we will
learn several different methods of proving things.
Propositions
A
proposition
is a statement which is either true or false.
These statements are all propositions:
(1)
√
3 is irrational.
(2) 1
+
1
=
5.
(3) Julius Caesar had 2 eggs for breakfast on his 10
th
birthday.
These statements are clearly not propositions:
(4) 2
+
2
(5)
x
2
+
3
x
=
5.
These statements aren’t propositions either (although some books say they are). Propositions should not
include fuzzy terms.
(6) Schwarzenegger often eats broccoli. (What is “often?”)
(7) George W. Bush is popular. (What is “popular?”)
Propositions may be joined together to form more complex statements. Let
P
,
Q
, and
R
be variables repre
senting propositions (for example,
P
could stand for “3 is odd”). The simplest propositional forms combine
variables using the connectives “and, or, and not.”
(1)
Conjunction
:
P
∧
Q
(“
P
and
Q
”). True only when both
P
and
Q
are true.
(2)
Disjunction
:
P
∨
Q
(“
P
or
Q
”). True when at least one of
P
and
Q
is true.
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 Fall '08
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