2/24/19 1 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 2 0 1 3
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INTRODUCTION TO INTEGERS · Matrices -Introduction A matrix is denoted by a bold capital letter and the elements within the matrix are denoted by lower case letters e.g. matrix [A]
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• 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
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