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Section 1.4: Properties of Integers

Examples {7}

INTRODUCTION TO INTEGERS

• Integers are positive and negative numbers.•…, -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, …

• Each negative number is paired with a positive number the same distance from 0 on a number line.

-3

-2

-1

20 1 3

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Integers

• Integers are the whole numbers and their opposites (no decimal values!)

• Example: -3 is an integer• Example: 4 is an integer• Example: 7.3 is not an integer

“Operators” & “Terms”…

12 • -5 + -3 • -6

Terms Operators

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Divisibility:

An integer a divides b (written “a|b”)

if and only if there exists an

Integer c such that c*a = b.

Primes:

A natural number p ≥ 2 such that

among all the numbers 1,2…p

only 1 and p divide p.

(a mod n) means the remainder when a is divided by n.

a mod n = r ⇔ a = dn + r for some integer d

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Greatest Common Divisor:GCD(x,y) = greatest k ≥ 1 s.t. k|x and k|y.

Least Common Multiple:LCM(x,y) = smallest k ≥ 1 s.t. x|k and y|k.

You can useMAX(a,b) + MIN(a,b) = a+bapplied appropriately to the factorizations of x and y to prove the above fact…

Fact:GCD(x,y) × LCM(x,y) = x × y

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4) Find the GCF of 42 and 60.

What prime factors do the numbers have in common?

Multiply those numbers.The GCF is 2 • 3 = 6

6 is the largest number that can go into 42 and 60!

42 = 2 • 3 • 760 = 2 • 2 • 3 • 5

5) Find the GCF of 40a2b and 48ab4.40a2b = 2 • 2 • 2 • 5 • a • a • b48ab4 = 2 • 2 • 2 • 2 • 3 • a • b • b • b • b

What do they have in common?Multiply the factors together.

GCF = 8ab

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What is the GCF of 48 and 64?

1. 22. 43. 84. 16

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Section 1.5: Matrices

Examples {12,13}

Matrices - IntroductionDefinition:A matrix is a set or group of numbers arranged in a square or rectangular array enclosed by two bracketsProperties:• A specified number of rows and a specified number of columns• Two numbers (rows x columns) describe the dimensions or size of the matrix.

Examples: 3x3 matrix2x4 matrix1x2 matrix

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Matrices - IntroductionA matrix is denoted by a bold capital letter and the elements within the matrix are denoted by lower case letters e.g. matrix [A] with elements aij

i goes from 1 to mj goes from 1 to n

Amxn=

Matrices - IntroductionTYPES OF MATRICES

1. Column matrix or vector:The number of rows may be any integer but the number of

columns is always 1

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Matrices - IntroductionTYPES OF MATRICES

2. Row matrix or vectorAny number of columns but only one row

Matrices - IntroductionTYPES OF MATRICES

3. Rectangular matrixContains more than one element and number of rows is not equal to the number of columns

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Matrices - IntroductionTYPES OF MATRICES

4. Square matrixThe number of rows is equal to the number of columns(a square matrix A has an order of m)

m x m

The principal or main diagonal of a square matrix is composed of all elements aij for which i=j

Zero-one matrices

Definition: A matrix all of whose entries are either 0 or 1 is called a zero-one matrix. (These will be used in Chapters 9 and 10.)Algorithms operating on discrete structures represented by zero-one matrices are based on Boolean arithmetic defined by the following Boolean operations:

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Zero-One (Boolean) Matrix

Definition:Entries are Boolean values (0 and 1)Operations are also Boolean

úûù

êëé= 010

101A úûù

êëé= 011

010B

úûù

êëé=úû

ùêëé

ÚÚÚÚÚÚ=Ú 011

111001110011001BA

úûù

êëé=úû

ùêëé

ÙÙÙÙÙÙ=Ù 010

000001110011001BA

Matrix join.• A Ú B = [ai,j Ú bi,j]

Matrix meet.• A Ù B = [ai,j Ù bi,j]

Example:

Zero-One (Boolean) Matrix

úúû

ù

êêë

é=

011001

A úûù

êëé= 110

011Búúû

ù

êêë

é=

011110011

BA!

Matrix multiplication: Am´k and Bk´n• the product is a Zero-One matrix, denoted AoB = Cm´n

• cij = (ai1 Ù b1j) Ú (ai2 Ù b2i) Ú … Ú (aik Ù bkj).Example:

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Joins and meets of zero-one matrices

Example: Find the join and meet of the zero-one matricesSolution:

The join of A and B is

The meet of A and B is

Illustration of matrix multiplication

The Product of A = [aij]andB = [bij]

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Boolean product of zero-one matrices

Definition: Let A = [aij]beanm × k zero-onematrixandB = [bij]beak × n zero-onematrix.TheBoolean product ofA and B, denoted by A ⊙ B, is the m×n zero-one matrix with(i,j)-th entry

cij = (ai1 ∧b1j)∨ (ai2 ∧b2j) ∨…∨(aik ∧bkj).Example:Find the Boolean product of A and B, where

Boolean product of zero-one matrices

Solution: The Boolean product A⊙ B is given by

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End of Ch 1

Chapter 1: Fundamentals• Section 1.1: Sets and subsets

Examples {1,5,6,8,9,10,11}• Section 1.2: Operations on Sets

Examples {1,2,3,4,6,7}• Section 1.3: Sequences

Examples {1,2,3,4,5,6,7,12}• Section 1.4: Properties of Integers

Examples {7}• Section 1.5: MatricesExamples {12,13}