6.1-6.2 Discrete Probability
General Definitions, Terms & Properties of Probability Theory
Sample space: the set of possible outcomes of an experiment.
Event: a subset of the sample space.
The relations:
sE S
where
s : an outcome
E : an event
S : a sample
3.4-3.7: Integers, Primes, Modular Arithmetic, Euclidean algorithm, and the Chinese Remainder Theorem
The Division Algorithm: Let a be an integer and d a positive
integer. Then there are unique integer q and r, with
such that a dq r .
0r d ,
Definition: q
1.4. Nested Quantififiers
Statement
When True?
Negation
xy P x, y
P x, y is true for every pair x, y
xy P x, y
xy P x, y
For every x , there is a y for which
P x, y is true
xy P x, y
There is an x for which P x, y is true
xy P x, y
for every y
There
7.3 Linear Recurrence Relations
Objective: To introduce the notion of a generating function. To
show how generating functions can be used to solve counting
problems, to solve recurrence relations, and to prove identities.
Definition. The generating functi
7.2 Linear Recurrence Relations
Objective: To solve linear recurrence relations with constant
coefficients.
Linear homogeneous recurrence relation of degree k
constant coefficients
an c1an 1 c2 an 2 ck an k , ci , ck 0
Property of Linear Homogeneous Recur
7.1 Recurrence Relations
Objective: To show how counting problems can be modeled
using recurrence relations.
Recurrence Relation
A recurrence relation for the sequence an is an equation that
expresses an in terms of one or more of the previous terms
an1 ,
5.5 Generalized Permutations and Combinations
Permutations with Repetition
The number of r permutations of a set of n objects with
repetition allowed is n r .
Example: The number of strings of length r from the English
alphabet is n r .
Equivalent Model o
5.2 The Pigeonhole Principle and Ramsey Theory
Objective: To introduce the pigeonhole principle and show how
to use it in enumeration and in proofs; to learn simple examples
of Ramsey theory.
The Pigeonhole Principle
If k and k 1 or more pigeons are place
5.4 Binomial Coefficients (addendum)
The Binomial Theorem
x y
n
n
x n j y j , x, y , n
j 0 j
n
n
n
j 0
Corollary. 1 x x j
j
n
Remark: A generalization of the corollary is the Binomial
Theorem in calculus about infinite series:
1 x x j , x,
j 0 j
Exa
4.3 Recursive Definitions and Structural Induction
Objective: Understand how functions, sequences and sets can be
defined recursively, and how to use induction, including structural
induction, to prove properties of such entities.
Recursively Defined Func
4.1-4.2: Mathematical Induction
Principle of Mathematical Induction
P n : a proposition function of n .
Goal: to prove that P n is true for all n .
I. Basis Step: Verify that P 1 is true.
II. Inductive Step: Show that k , P k P k 1 is true.
Example. Let a
3.8: Matrices
Matrix:
A aij
mn
a11 a1n
.
a
m1 amn
Matrix Entries
The entries aij can be in a field, such as , , ,
or a ring, such as , n m 0,1,2, m 1 , with routine
addition and multiplication.
Or aij 0,1 , a Boolean ring, with Boolean addition and
3.1-3.2-3.3: Algorithms-Growth of Functions-Complexity
Algorithm: A finite set of precise instructions for performing a
computation or for solve a problem.
English description of algorithm <
pseudocode < programming language
General Properties of Algorith