Chapter 1
What is a Proof?
1.1
Mathematical Proofs
A proof is a method of establishing truth. What constitutes a proof differs among
elds.
Legal truth is decided by a jury based on allowable evidence presented at
trial.
Authoritative truth is specied
V. Adamchik
21-127: Concepts of Mathematics
Mathematical Induction
Victor Adamchik Fall of 2005
Lecture 1 (out of three)
Plan
1. The Principle of Mathematical Induction 2. Induction Examples
The Principle of Mathematical Induction
Suppose we have some sta
15-251 Assignment 10
Page 1 of 8
15-251 : Great Theoretical Ideas In Computer Science
Fall 2012
Assignment 10
Due: Thursday, Nov.29, 2012
Question:
1
2
Total
Points:
80
20
100
Score:
15-251 Assignment 10
Page 2 of 8
1. ACM
Life Cycle of a Program
Source C
15-251: Great Theoretical Ideas in Computer Science
Common Mistakes in Induction Proofs (draft!)
Consider the problem of proving that n 0, 1 + 2 + . . . + n =
Dene the statement Sn = 1 + 2 + . . . + n =
1
n(n+1)
.
2
Anupam Gupta
January 28, 2011
n(n+1)
2
Chapter 18
Introduction to Probability
Probability plays a key role in the sciences hard and social including com
puter science. Many algorithms rely on randomization. Investigating their cor
rectness and performance requires probability theory. Moreover,
15-251: Great Theoretical Ideas In Computer Science
Recitation Supplement - Lambda Calculus
Syntax
Lambda expressions are dened recursively as follows:
Variables: a, b, c, . . . are lambda expressions.
Lambdas: If e is a lambda expression then x.e is a la
Great Theoretical Ideas In Computer Science
Victor Adamchik
Lecture 2
CS 15-251
Carnegie Mellon University
Inductive Reasoning
American Banks in 2008
Domino Effect: Line up any
number of dominos in a row;
knock the first one over and
they will all fall
Ra
15-251: Great theoretical ideas in Computer Science
Carnegie Mellon University
Notes on group theory
October 2011
A. Gupta & V. Guruswami
Excerpts from Chapters 3, 5, 6 of
Abstract Algebra: Theory and Applications
by Thomas W. Judson
The textbook is avail
V. Adamchik
21-127: Concepts of Mathematics
Mathematical Induction
Victor Adamchik Fall of 2005
Lecture 2 (out of three)
Plan
1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal
Strong Induction
Fibonacci Numbers Fibonacc
V. Adamchik
21-127: Concepts of Mathematics
Mathematical Induction
Victor Adamchik Fall of 2005
Lecture 3 (out of three)
Plan
1. Recursive Definitions 2. Recursively Defined Sets 3. Program Correctness
Recursive Definitions
Sometimes it is easier to defin
A Tutorial Introduction to the Lambda Calculus
Ral Rojas u FU Berlin, WS-97/98
Abstract This paper is a short and painless introduction to the calculus. Originally developed in order to study some mathematical properties of eectively computable functions,
15-251: Great Theoretical Ideas in Computer Science
Fall 2012, Lecture 3
Venkat Guruswami
Axiomatic Systems & Logic
P, P Q
Q
1+1 = 2
In mathematics, sometimes your intuition
can be quite wrong.
do we need a proof?
So it really pays off to:
Formalize conce
15-251: Great Theoretical Ideas in Computer Science
Lecture 4
Recap: Axiomatic systems
Vocabulary
(Universe of
expressions)
Proofs
Theorems
in the
axiomatic
system
(whatever can be proved in finitely many steps using the
Axioms & deduction rules)
Recap: T
Probability Theory II
(Setting Expectations for
Computer Scientists)
Lecture 11
October 2, 2012
Review
Some useful
sample spaces
E[X+Y] = E[X] + E[Y]
1) A fair coin
3) Two independent bias-p coin tosses
sample space S = cfw_H, T
Pr(H) = , Pr(T) = .
sample
15-251: Great Theoretical Ideas in Computer Science
Lecture 19
Linear algebra is about vectors.
October 30, 2012
Linear Algebra
Concretely, vectors look like this:
They are lists of numbers.
fig. by Peter Dodds
# of numbers, m, is called the dimension.
In
Great Theoretical Ideas in CS
V. Adamchik
CS 15-251
Fall 2012
Plan
Carnegie Mellon University
Graphs Meet Linear Algebra
Lights Out
Lights Out in F2
Let xi represent the number of times a
square i is touched. The number of
times a square changes the state
15-251
Great Theoretical Ideas in
Computer Science
November 6, 2012
Election Day!
Danny Sleator (guest lecture)
How should we
vote?
Plan for Today:
1: Voting
2: Electoral College
3: Voting a topic off the final
Part 1: The System is Broken
Consider the 20