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Discrete Math

Instructor’s Information:Name: Dr. Najah Al-shanablehOffice hours: Sunday and Tuesday : 9:30 -11:00 amMonday and Wednesday: 11:00 am – 12:00 pmE-Mail: najah2746@aabu.edu.joOffice Tel.: 3395

Sunday, Tuesday

Class Hall Time

02 201 IT 11-12:30

Monday, Wednesday

Class Hall Time

03 104 IT 12:30-2

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Course Description

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Course Description

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The Course Plan

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Textbook:

Discrete Mathematical Structures, B. Kolman , RC. Busby and SC Ross, Prentice Hall, 6th Edition, 2008

Grading:

Mid-term Exams (2): 50% Final Exam: 50%

The instructor encourages everyone to participate in class activities, discussions, and respond to questions from other students and complete in/out-class writing assignments.

Tentative Course Outline/Schedule:

No. Topic Hours

No. Topic Hours

1 Sets & Sequences 3 8 Relations & Diagraphs 32 Division in the Integer &

Matrices3 9 Equivalence Relations &

Operations on Relations3

3 Propositions & Logical Operations

3 10 Functions & Functions for Computer Science

3

4 Conditional Statements & Methods of Proof

3 11 Labeled, Searching & Minimal Spanning Trees

3

5 Mathematical Induction 3 12 Euler & Hamiltonian Paths/Circuits

3

6 Permutations & Combinations

3 13 Finite State Machine 3

7 Elements of Probability 3 14 Experiments in Discrete Mathematics

3

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ةعطقتملا تایضایرلاChapter 1: Fundamentals• Section 1.1: Examples {1,5,6,8,9,10,11}• Section 1.2: Examples {1,2,3,4,6,7}• Section 1.3: Examples {1,2,3,4,5,6,7,12}• Section 1.4: Examples {7}• Section 1.5: Examples {12,13}Chapter 2: Logic• Section 2.1: Examples {1,2,3,4,5,}• Section 2.2: Examples {1,2,3,4}• Section 2.4: Examples {1,2}Chapter 3: Counting• Section 3.1: Examples {8,9,10}• Section 3.2: Examples {3}Chapter 4: Relations & Digraphs• Section 4.1: Examples {1,2,6}• Section 4.2: Examples {1,2,3,4,10,11,18,19,22,23,24}• Section 4.3: Examples {5,6}• Section 4.4: Examples {1(c),4,10}• Section 4.5: Examples {2}• Section 4.7: Examples {1,3,4,5,6,10,12,13}Chapter 5: Functions• Section 5.1: Examples {1,2,912,14}• Section 5.4: Examples {2,3,4,5,6,7,9,10}Chapter 7: Trees• Section 7.1: Theorem 1• Section 7.2: Examples {3,4}• Section 7.3: Examples {1,2,3,4}• Section 7.5: Examples {1,2,5,6}Chapter 8: Topics in Graph Theory• Section 8.1: Examples {1,2,3}• Section 8.2: Examples {1,2,4}• Section 8.3: Examples {1}Chapter 10: Languages & FSMs• Section 10.3: Examples {1,2,3,4}• Section 10.4: Examples {1}

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1.1 Sets and Subsets

• Section 1.1: Examples {1,5,6,8,9,10,11}

Introduction

• A set is a collection of objects.• The objects in a set are called elements of the set.

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Ways to define sets• Explicitly: {John, Paul, George, Ringo}• Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…}• Set builder: { x : x is prime }, { x | x is odd }. • In general { x : P(x)}, where P(x) is some predicate.

We read “the set of all x such that P(x)”

Set – Builder Notation

• When it is not convenient to list all the elements of a set, we use a notation the employs the rules in which an element is a member of the set. This is called set – builder notation. • V = { people | citizens registered to vote in Maricopa County}• A = {x | x > 5} = This is the set A that has all real numbers greater than

5. • The symbol | is read as such that.

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Set properties 1

• Order does not matter• We often write them in order because it is easier for humans to understand it

that way• {1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1}

• Sets are notated with curly brackets

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Set properties 2

• Sets do not have duplicate elements• Consider the set of vowels in the alphabet.

• It makes no sense to list them as {a, a, a, e, i, o, o, o, o, o, u}• What we really want is just {a, e, i, o, u}

• Consider the list of students in this class• Again, it does not make sense to list somebody twice

• Note that a list is like a set, but order does matter and duplicate elements are allowed• We won’t be studying lists much in this class

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Specifying a set 1

• Sets are usually represented by a capital letter (A, B, S, etc.)

• Elements are usually represented by an italic lower-case letter (a, x, y, etc.)

• Easiest way to specify a set is to list all the elements: A = {1, 2, 3, 4, 5}• Not always feasible for large or infinite sets

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Specifying a set 2• Can use an ellipsis (…): B = {0, 1, 2, 3, …}• Can cause confusion. Consider the set C = {3, 5, 7, …}. What comes next?• If the set is all odd integers greater than 2, it is 9• If the set is all prime numbers greater than 2, it is 11

• Can use set-builder notation• D = {x | x is prime and x > 2}• E = {x | x is odd and x > 2}• The vertical bar means “such that”• Thus, set D is read (in English) as: “all elements x such that x is prime and x is

greater than 2”

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Specifying a set 3

• A set is said to “contain” the various “members” or “elements” that make up the set• If an element a is a member of (or an element of) a set S, we use then

notation a Î S• 4 Î {1, 2, 3, 4}

• If an element is not a member of (or an element of) a set S, we use the notation a Ï S• 7 Ï {1, 2, 3, 4}• Virginia Ï {1, 2, 3, 4}

Example 1

• Let A ={ 1,3,5,7]. Then 1 ϵ A, 3 ϵ A, but 2 Ï A.

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Special Sets of Numbers• N = The set of natural numbers. The whole numbers from 1 upwards.

(Or from 0 upwards in some fields of mathematics= {1, 2, 3, …}.

• W = The set of whole numbers.={0, 1, 2, 3, …}

• Z = The set of integers.= { …, -3, -2, -1, 0, 1, 2, 3, …}

• Q = The set of rational numbers.={x| x=p/q, where p and q are elements of Z and

q ≠ 0 }• H = The set of irrational numbers.• R = The set of real numbers.• C = The set of complex numbers.

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Universal Set and Subsets• The Universal Set denoted by U is the set of all

possible elements used in a problem.• When every element of one set is also an element of

another set, we say the first set is a subset. • Example A={1, 2, 3, 4, 5} and B={2, 3}

We say that B is a subset of A. The notation we use is B A.• Let S={1,2,3}, list all the subsets of S.• The subsets of S are , {1}, {2}, {3}, {1,2}, {1,3}, {2,3},

{1,2,3}.

Í

Æ

The Empty Set• The empty set is a special set. It contains no

elements. It is usually denoted as { } or.

• The empty set is always considered a subset of any set.• Do not be confused by this question:• Is this set {0} empty? • It is not empty! It contains the element zero.

Æ

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Intersection of sets

• When an element of a set belongs to two or more sets we say the sets will intersect.• The intersection of a set A and a set B is denoted by A ∩ B. • A ∩ B = {x| x is in A and x is in B}• Note the usage of and. This is similar to conjunction. A ^ B.• Example A={1, 3, 5, 7, 9} and B={1, 2, 3, 4, 5}• Then A ∩ B = {1, 3, 5}. Note that 1, 3, 5 are in both A and B.

Mutually Exclusive Sets

• We say two sets A and B are mutually exclusive if A ∩ B = . • Think of this as two events that can not happen at the same time.

Æ

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Union of sets

• The union of two sets A, B is denoted by A U B.• A U B = {x| x is in A or x is in B}• Note the usage of or. This is similar to disjunction A v B. • Using the set A and the set B from the previous slide, then the union

of A, B is A U B = {1, 2, 3, 4, 5, 7, 9}.• The elements of the union are in A or in B or in both. If elements are

in both sets, we do not repeat them.

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Examples 5 & 6

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Examples 7 & 8

Examples 9 & 10

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Cardinality

If S is finite, then the cardinality of S, |S|, is the number of distinct elements in S.

If S = {1,2,3} |S| = 3.

If S = {3,3,3,3,3}

If S = Æ

If S = { Æ, {Æ}, {Æ,{Æ}} }

|S| = 1.

|S| = 0.

|S| = 3.

If S = {0,1,2,3,…}, |S| is infinite. (more on this later)

Power setsIf S is a set, then the power set of S is

P(S) = 2S = { x : x Í S }.

If S = {a}

If S = {a,b}

If S = Æ

If S = {Æ,{Æ}}

We say, “P(S) is the set of all subsets of S.”

2S = {Æ, {a}}.

2S = {Æ, {a}, {b}, {a,b}}.2S = {Æ}.

2S = {Æ, {Æ}, {{Æ}}, {Æ,{Æ}}}.

Fact: if S is finite, |2S| = 2|S|. (if |S| = n, |2S| = 2n)

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Example 11