Conditional probability
11/10/11
Example: RE = roll of a fair 6-face die. So the sample space = cfw_1, 2,394.56. Let A = prime number occurs = cfw_2, 3.5. P(A) = 3/6 = 1/2 and B = even number occurs = cfw_2,4,6. P(B) = 3/6 = 1/2. In a roll of the die, it
Modular Arithmetic
Arithmetic of remainders resulting from divisions of integers by a positive integer.
8/30/11
Theorem 4.
Theorem 3.
Theorem 5.
Fact: a (a mod m) (mod m). Why? By the division algorithm, there exist two unique integers q and r such that a
Number Theory
8/25/11
Division
Example: Does 9 divide 36? yes 9 | 36 True or False: 11 | 120? False Example: Let n and d be positive integers. How many positive integers not exceeding n are divisible by d?
x is the smallest integer x. x is the largest int
Matrices
9/8/11
Special matrices
If a matrix has only one row, then it is a row vector.
If a matrix has only one column, then it is a column vector.
is a 4x1 matrix or a column vector.
A 1x1 matrix is a scalar.
is a 1x1 matrix or a scalar.
A null matrix h
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Primes
Definition: Let P be a positive integer greater than 1. p is a prime number if the only positive factors of p are 1 and p. p is a composite number if it has a positive factor other than 1 and itself.
9/1/11
Theorem 1. Fundamental Theorem of Arithme
Number Systems
9/6/11
Decimal number system
A position's weight is rk, where k is the number of positions from the decimal point. To the left of the decimal point, k takes values 0, 1, 2. , and to the right, it takes values -1,-2,.
Binary number system
Sp
Principle of Inclusion-Exclusion (PIE)
10/27/11
Generalized Principle of Inclusion and Exclusion (PIE)
Applications of PIE
Example, Section 8.6, Page 560
Examplel 1, Section 8.6
Not solved in the class
15, Section 8.5 Count all 10 digit permutations that
Probability
11/8/11
Event: a subset of the sample space , satisfying certain axioms.
Example 5. A system consists of two subsystems: one with four components and the other with three components. We are interested in only working components of the system.
10/25/11
Indistinguishable objects into distinguishable boxes (IODB) Similar to combinations with repetitions. For example, consider the previously solved science fair problem. Three schools - A, B and C - are competing for a grand prize in a science fair
Generalized Permutations and Combinations
10/20/11
Counting with n distinct objects
Combinations with repetition Example: Three schools - A, B and C - are competing for a grand prize in a science fair competition. There are two judges. Each judge, anonymo
Binomial Coefficients
10/18/11
Binomial Theorem
Binomial Theorem
Combinatorial proof:
Example
Examples
Pascal's Triangle
Pascal's Identity
Combinatorial argument:
have 'a' in the k elements chosen
Do not have the 'a' in the k elements chosen
Algebraic pro
10/11/11
33. Consider strings of eight English letters. Count (d) strings that start with a vowel, letters cannot be repeated
(g) strings that start with x and contain at least one vowel; letters can be repeated = number of strings that start with x - num
Permutations
10/13/11
An ordered arrangement of r elements out of n distinct elements is called an r-permutation.
For the above problem, we counted 3-permutations of 3 distinct elements:
Example: If the number of available students is 5 and the line shoul
Pigeonhole Principle (PhP)
[KR, Section 6.2]
10/6/11
Theorem 1. If k is a positive integer and k+1or more objects are to be placed in k boxes, then at least one box contains at least two objects.
Example 3 The possible scores in a test are 0, 1, ., 100. W
Combinatorics
Basic Counting Principles [KR, Section 6.1] Product Rule
10/4/11
Example 0: What is the number of ways to have a lunch if a lunch consists of a sandwich and a drink? There are five distict types of sandwiches and three distinct types of drin
Eigenvalues and Eigenvectors
9/22/11
Consider an equation of the form Ax = x, where A=(ai,j)nxn a matrix of knowns, x=(xi)nx1 a vector of unkowns, and is an unkown scalar.
If the equation is satisfied for x other than the null vector, then each such x is
9/20/11
9. Show that A + (B+C) = (A+B) + C, where A, B and C are matrices of order mxn.
11. If AB and BA are defined, what can you say regarding the sizes of A and B?
12a. Show that (A + B)C = AC + BC, where A, B and C are matrices and the sum A+B and pro
Rank of a matrix
9/15/11
Rank of A is the size of the largest square submatrix of A whose determinant is nonzero.
Rank of A is 2. All you need to show is there is a 2x2 submatrix of A that is nonsingular - determinant is nonzero.
Example:
Inverse of a mat
9/13/11
Matrix multiplication Suppose A is an mxp matrix and B a qxn matrix. The matrix multiplication AB is defined if p=q. The result is a matrix of order mxn. The (i,j)th entry of the result matrix is the dot product of row i of A and column j of B.
Ex