Introduction to Discrete Mathematics Ch3

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Chapter 3The Fundamentals: Algorithms, the integers, and matrices

Section3.1: Algorithms

Definition1: an algorithm is a finite set of precise instructions for performing acomputation or for solving a problem.

Example1: page168: describe an algorithm for finding the maximum value in a finitesequence of integers. Answer: we can specify a procedure for solving this problem in several ways. One

method is simply to use the English language to describe the sequence of steps used.1- Set the temporary maximum equal to first integer in the sequence2- Compare the next integer in the sequence to the temporary maximum, if it is larger than the temporary max., set the temporary maximum equal to this integer.3- Repeat the previous step if there are more integers in the sequence.4- Stop when there are no integers left in the sequence. The temporary maximum at this

point is the largest integer in the sequence.

Remark: 1- An algorithm can also be described using a computer language.2- Pseudo code: provides an intermediate step between an English language

description of an algorithm and an implementation of this algorithm in a programminglanguage.



3- The pseudo code used in this book is loosely used on programming language Pascal.4- A pseudo code description of the algorithm for finding the maximum element in a finite

sequence follows:

Read searching Algorithms-sorting (The bubble sort- The intersection sort- Greedy Algorithms).

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Section 3.4: The integers and divisionNumber theory: the part of mathematics involving integers and their properties belongs

to the branch of mathematics called number theory.Definition: if a and b are integers with a�0 we say that a divides b if there is an integer c

such that b = ac. (Note that a is a factor of b)Remark: 1- We denote a divides b by ab

2- We denote a doesn¶t divide b by a »bExample1: determine whether 37 and whether 312answer: 3 does not divide 7 and 312 since 12 = 3.4

Example2: let n and d be positive integers. How many positive integers not exceeding nare divisible by d? Answer: The positive integers divisible by d is of the form kd. So kd is not exceeding n.

So . Therefore there are n/d positive integers not exceedingn that are divisible by d. k

nd nd k ee 00



Theorem 1:Let a,b and c be integers. Theni) If ab and ac, then a(b+c)ii) If ab then abc for all integers c;

iii) If ab and bc, then ac

Proof:

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Corollary: If a,b, and c are integers such that ab and a/c, then a mb+nc whenever mand n are integers.

Proof: Since ab by theorem(1) ii ambSince ac by theorem (1) ii anc

So by theorem (1) I amb+nc

Theorem 2: The Division AlgorithmLet a, b, and c are integers and d a positive integer. Then there are unique q and r, with

o � r < d, such that a = dq + r.

Example: let a = 17d = 3 then 17 = 3.5 + 2

Definition: In the theorem 2, d is called the divisor, a is called the dividend, q is called the

quotient, and r is called the remainder.q = a div d, r = a mod d.

Example3: what are q and r when 101 is divided by 11?Solution: 101 = 9(11)+2

q = 9 , r =2So 9 = 101 div 11 2 = 101 mod 11



Example4: what are the quotient and remainder when -11 is divided by 3? Answer: -11 = 3(-4) + 1so -4= -11 div 3

1 = -11 mod 3

Note that r can¶t be negative. 0 � r < 3

Remark: the integer a is divisible by the integer d if and only if the remainder is zero.

Modular Arithmetic:

Definition3: If a and b are integers and m is a positive integer, then a is congruent to bmodulo m if m divides a-b. We use notation a b (mod m) to indicate that acongruent to b modulo m.

Theorem3: let a and b be integers, and let m be a positive integer. Then a b (mod m)

if and only if a mod m = b mod m.Ex5: 1- determine whether 17 is congruent to 5 modulo 62- Determine whether 24 is congruent to 14 modulo 6

Solution: 17 ± 5 = 12, 612 so 17 5 (mod 6)24 ± 14 = 10, 6»10 so 24 is not congruent to 14 (mod 6)

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Theorem4: let m be a positive integer. The integers a and b are congruent modulo m iff there is an integer k such that a = b + km

Proof: suppose a b mod (m) m / a-b a-b=km

a=b+kmConversely: suppose a = b + km k is an integer

a ± b = km m / a ± b ( definition of divisibility) a b ( mod m)

Theorem5: let m be a positive integer. If a b (mod m) and c d (mod m), thena + c b + d (mod m) and ac bd (mod m).Proof: since a b (mod m) m / a ± b a = b + mSince c d (mod m) m / c- d c = d + ma + c = (b + d) + m (a + c) ± (b + d) =

m / (a + c) ± (b + d) a + c b + d mod (m)a . c =

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Example6: 1- Show that 18 3 (mod 5), by using theorem 5:Solution: 18 = 7 + 11, 3 = 2 + 1 7 2 mod (5)

11 1 mod (5)18 = 7 + 11 2 + 1 = 3 (mod 5)

so 18 3 (mod 5)2- Show that 77 2 (mod 5) by using theorem 577 = (7)(11) (2)(1) (mod 5)77 2 (mod 5)

Corollary 2: let m be a positive integer and let a and b be integers. Then(a + b) mod m = ((a mod m) + (b mod m)) mod m And ab (a mod m) (b mod m) (mod m)

Application of Congruences

Read example7. Hashing FunctionsRead example8Remark on example8:1- Pseudorandom Numbers: numbers generated by systematic methods are not truly

random, they are called Pseudorandom Numbers.

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2- we generate a sequence of pseudorandom numbers with for all nby using the congruence

Cryptology: is the study of secret messages.Caesar encryption: is the process of making a message secret by shifting each letter

forward. Eg. B sends to E, X sends to FCaesar¶s encryption method can be represented by the function f that assigns to the

nonnegative integer p, p � 25, the integer f(p) is the set {0,1,««,25} withf(p) = (p+3) mod 26 note that in this process we replace each letter by an integer from 0to 25, based on it¶s position for example replace A by 0, K by 10 and so on.

Example9: what is the secret message produced from the message ³Meet you in thepark´ using the Caesar Cipher?

Solution: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 251) Replace the letters in the message with numbers

M e e t y o u I n t h e p a r k12 4 4 19 24 14 20 8 13 19 7 4 15 0 17 10

_ an x m xn e0

m xand mcma

with seed xmca x x nn

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2) Replace each of these numbers ply f(p)=(p+3)mod 2615 7 7 22 1 17 23 11 16 22 10 7 18 3 20 13

3) Translating this back to letters produces encrypted message

15 7 7 22 1 17 23 11 16 22 10 7 18 3 20 13P H H W B R X L Q W K H S D U N

Decryption: the process of determining the original message from the encryptedmessage is call decryption

Remark: instead of shifting each letter by 3, we can shift each letter by k so thatf(p)=(p+k) mod26 (called shift cipher)

Example: Decrypt the message using the Caesar Cipher:E O X H M H D C V

4 14 23 7 12 7 3 16 21

1 11 20 4 9 4 0 13 18 ( f(p)=p-3 )

B L U E J E A N S

Do example10 page 208



Using f(p)=(p+3)(mod26)Decrypt the message

G R Q R W S D V V

6 17 16 17 22 18 3 21 213 14 13 14 19 15 0 18 18³D O N O T P A S S´

Encrypt the message ³ how are you´

H O W A R E Y O U7 14 22 0 17 4 24 14 2010 17 25 3 20 7 1 17 23K R Z D M H B R X



Section 3.5Primes and Greatest Common Divisors

A prime: is an integer greater than 1 that is divisible only by 1 and by itself.Definition: a positive integer p greater than 1 is called prime if the only positive factors of

p are 1 and p.

Remark: 1- A positive integer that is greater than 1 and is not prime called composite.2- The integer n is composite iff there exists an integer a such that an and 1 < a < n.

Examples of prime numbers:2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89 and 97

Theorem1: the fundamental theorem of Arithmetic. Every positive integer greater than 1Can be written uniquely as a prime or as the product of two or more primes where the

Prime factors are written in order of non decreasing size.

Example2: Find the prime factorization of 100,641,999, and 1024.Solution: 100 = 2.2.5.5

= 2².5²

641 = 641 X 1 (prime)



999 = 3³ . 37

1024 = 2

Theorem2: If n composite integer, then n has a prime divisor less than or equal to

Example: Show that 101 is primeSolution: The only prime not exceeding ¥101 are 2,3,5 and 7. But ¥101 is not divisible by

2,3,5, or 7 it follows that 101 is prime.

Example4: Find the prime factorization of 7007.Solution: start with 2 then 3 then 5 . None of them divides 7007Try 7, we get 7007 = 7² X 11 X 13

Theorem3: There are infinitely many primesMersenne primes: The largest prime known has been an integer of special form 2 ± 1

10

p

n



Example5: page 212

Theorem4: The prime number theorem. The ratio of the number of primes are notexceeding x and x / lnx approaches 1 as x grows without bound.

Conjectures and open problems about primes.1- Consider the polynomial f(n) = n² - n + 41. This polynomial has the property that f(n) is

prime for all positive integers not exceeding 40.Eg: f(1) = 41 f(3) = 47 f(2) = 43 f(4) = 53 and so on.2- Every odd integer n, n > 5 is the sum of 3 primes

Every even integer n, n > 2 is the sum of 2 primesEg: 4 = 2 + 2, 6 = 3 + 3, 8 + 5 + 3, 10 = 7 + 3 and so on3- There are infinitely many primes of the form n² + 1 where n is a positive integer.Eg: 5 = 2² + 1, 17 = 4² + 1, 37 = 6² + 1 and so on4- The twin prime conjecture: Twin primes are primes that differ by 2 such as 3 and 5, 5

and 7, 11 and 13, 17 and 19.P + 2 where p is prime and p + 2 is prime or the product of two primes .

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Greatest Common Divisors and Least Common MultiplesDefinition2: let a and b be integers, not both zero. The largest integer d such that da

and db is called the greatest common divisor of a and b denoted by gcd (a,b).Remark: One way to find the greatest common divisor of two integers is to find all the

positive common divisors of both integers and then take the largest divisior.Example10 page 215What is the greatest common divisor of 24 and 36 ?Solution: 24:1,2,3,4,6,8,12,24 (factors of 24)

36: 1,2,3,4,6,9,12,16,36 (factors of 36)

g.c.d(24,36) = 12Example11: Find g.c.d (17,22)Solution: 17: 1, 17

22: 1, 2,11,22g.c.d (17, 22) = 1

Definition3: The integers a and b are relatively prime if their greatest common divisor is1.Example12: Since g.c.d (17,22) = 1 it follows that the integers 17 and 22 are relativelyprime.

Definition4:

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Example13: Determine whether the following integers are pair wise relatively prime1- 10, 17, 212- 10, 19, and 24Solution: 1- g.c.d(10,17)=1

g.c.d(17,21)=1g.c.d(10,21)=1

We calculate that 10,17,21 are pair wise relatively prime.2- g.c.d(10,19)=1

g.c.d(19,24)=1

g.c.d(10,24)=2>1So 10,19 and 24 are not pair wise relatively prime.

Remark: Another way to find the prime g.c.d if 2 integers is to use the primefactorizations of these integers suppose

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Example14: find the g.c.d (120,500) (using method 2, prime factorization)

120 = 2³.3.5

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500 = 2².5³

10 10

5 2 5 2

least common multipleDefinition5: The least common multiple of the positive integers a and b is the smallestPositive integer that is divisible by both a and b it is denoted by Lcm(a,b)Remark: lcm(a,b) =

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Example: What is the least common multiple of Answer:

Theorem5: let a and b be positive integers then ab = gcd (a,b) . Lcm (a,b)

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Section 3.6: Integers and Algorithms

Representations of Integers:Theorem(1): (The base b expansion of n).

Let b be a positive integer greater than1. Then if n is a positive integer, it can beexpressed uniquely in the form

Where k is a non negative integer, are non negative integers less thanb and

Example: 965 = 9 X 10² + 6 X 10 + 5 X 10Û the base here is 10Remark: (1) In theorem 1: The base b expansion of n is denoted byExample: (245) represents 2.8²+4.8+5=165 the base here is 8

(2) Choosing 2 as the base gives binary expansions of integers.(3) In binary notation each digit is either a0 or a1. So the binary expansions of

an integer is just a bit string.

Example1: What is the decimal expansion of the integer that has (101011111) as itsbinary expansion?Solution:

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This process terminates when we obtain a quotient equal to zeroExample3 page 220: Find the base 8, or octal, expansion of (12345)Solution: Divide 12345 by 8 12345 = 8.1543 + 1

Divide 1543 by 8 1543 = 8.192 + 7

192 = 8.24 + 024 = 8.3 + 03 = 8.0 + 3

Because the remainders are the digits of the base 8 expansion of 12345, it follows that

Example4: Find the hexadecimal expansion of (177130)Solution: 177130 = 16.11070 +10

11070 = 16.691 + 14691 = 16.43 + 3

43 = 16.2 + 112 = 16.0 + 2

10

810 3007112345 !

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Example: Find the binary expansion of (241)Solution: 241 = 2.120 + 1

120 = 2.60 + 060 = 2.30 + 0

30 = 2.15 + 015 = 2.7 + 17 = 2.3 + 13 = 2.1 + 11 = 2.0 + 1

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Section 3.8: Matrices

Definition1: A matrices is a rectangular array of numbers. A matrix with rows m and ncolumn is called an mXn matrix.

Remarks: 1- A matrix with the same number of rows as column is called square.2- The matrices are equal if they have the same number of rows and the same number

of columns and the corresponding entries in every position are equal.Example: The matrix is a 3X2 matrix

The matrix is a square matrix 2X2 matrixDefinition: Addition of matricesLet be mXn matrices. The sum of A and B, denoted by

A+B is the mXn matrix that has as its (i,j) the element. A+B=Remark: Addition of matrices id defined if the matrices are of same sizeExample: Add

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Definition: multiplication of matricesLet A be an mXk matrix and B be a kXn matrix. The product of A and B, denoted by AB

is the mXn matrix with its (i,j) the entry equal to the sum of the product of thecorresponding elements from the i th row of A and the j th column of B.

Remark: Multiplication of matrices is not commutative.Example:

Find AB if it is defined A is 4X3 B is 3X2 so AB is defined and is 4X2 matrix.

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Definition: A square matrix A is called symmetric if A=A

Example: The matrix is symmetric.

Definition: Zero-one Matrix A matrix with entries that are either 0 or 1 is called a zero- one matrix

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Introduction to Discrete Mathematics Ch3

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