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 relatio
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 intege
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
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
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 c
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
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
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 Pige
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 c
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 prov
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. Inductiv
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.
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 <
pseudocod