Discrete Math 2/10/19

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1 Discrete Math Instructor s Information: Name: Dr. Najah Al-shanableh Office hours: Sunday and Tuesday : 9:30-11:00 am Monday and Wednesday: 11:00 am 12:00 pm najah2746@aabu.edu.jo Office Tel.: 3395 Sunday, Tuesday Class Hall Time IT 11-12:30 Monday, Wednesday Class Hall Time IT 12:30-2 1

2 Course Description Click Here Course Description Click Here 2

3 The Course Plan 3

4 Textbook: Discrete Mathematical Structures, B. Kolman, RC. Busby and SC Ross, Prentice Hall, 6 th 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 Hour s No. Topic Hour s 1 Sets & Sequences 3 8 Relations & Diagraphs 3 2 Division in the Integer & 3 9 Equivalence Relations & 3 Matrices Operations on Relations 3 Propositions & Logical 3 10 Functions & Functions for 3 Operations Computer Science 4 Conditional Statements & 3 11 Labeled, Searching & Minimal 3 Methods of Proof Spanning Trees 5 Mathematical Induction 3 12 Euler & Hamiltonian 3 Paths/Circuits 6 Permutations & 3 13 Finite State Machine 3 Combinations 7 Elements of Probability 3 14 Experiments in Discrete Mathematics 3 4

5 ة$ ا مL ا مL ه ا$ ا ل ر ی ا ض ی ا ت ا ل م ت ق ط ع ة Chapter 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} ا ل # ا د ة (3 ( ت ع # $ & ا لا ع + ا ل ا ل # ا ل / ة م 2 ا ل ف ا ت ت ا د ی $ / ة ت ع & ض ا ل 9 ا ل : ا ل ; < ی & ت=: أ?ا مAها ل ل ع ق F E ا ت ا ل # ا د ی $ / ة ا ل + A E G ص ع ل / ه ا ف ي ه ; ا ا ل L A ا م :- أ- الام# Aاع ال+Rب& عU حWEر ال+ Yاض&ات أو الRروس أو عU الEاج$ات الا خ&_ ا ل # ي ت ق W ي ا لا ن + L ة a ا ل + اE b ع ل / ه ا و d ل ت g f& Y ع ل ى ه ; ا ا لا م # A ا ع. ب- ا ل غ l ف ي ا لا خ # $ ا ر أ و ا لا م # Y ا ن أ و ا لا ش # &ا ك أ و ا ل p &وع ف / ه وا لا خ لا ل ب A ا لا خ # $ ا ر أ و ا لا م # Y ا ن أ و ا ل ه R و ء ا ل E ا ج : ت E ا ف & ه ف / ه. أ- ا لا خ لا ل a ا ل A أ و ا لا ن $ W ا v ا ل ; < ت ق # W / ه ا ل + Y ا ض ا & ت أ و ا ل R A و ا ت أ و ا لا ن 9 p ة ا ل # ي ت ق ا م د ا خ ل ا ل w ا م ع ة. ب- أ < إ ه ا ن ة أ و إ س ا ء ة ی & ت = ا ل 9 ا ل : a Y { ع W E ه / ة ا ل # R ر } f أ و أ < م U ا ل ع ا م ل / U أ و ا ل 9 ل $ ة ف ي ا ل w ا م ع ة. 5

6 ي;.h ه ا د ة$ ة$ م ا ت$ ا 2 ة 2 ة$ الغHJ اF ل هN ةO ع ة :-C ةA ا ل I H ا مa ل 3 تS ا ل # ا د ة (4 ( ت " $ د ا ل ع ق +, ا ت ع ل ى ا ل 3 2 ا ل ف ا ت ا ل 5 ا د ی 9 8 ة ل ل ; ا ل < ا ل + ا ر د ة ف ي ا ل 2 ا د ة (3 ( م E ه G ا ا ل I H ا م ع ل ى ا ل " H + ا ل 5 ا ل ي :-. ا ل 9 8 H 5 ه ا ل 3 أ- ب- ا لا خ ا S ج م E ق ا ع ة ا ل $ 5 ر W V و ا س $ 5 ع ا ء ا لا م E ا ل \ ا م ع ي ع $ H ا ل ] ةS و ر لا خ ا S ج ه. ا ل ; ا ل < ا ل " نS م ا م E ح ] ر+ a ع b أ و c ل م " ا ض ا ا ل 2 +ا د ا ل 5 ي f ج- أ ث H ا ء ت $ ر V ا ل 5 ي f ق $ م ه ا م Sف k أ و أ ك m S م E م E ا لا س 5 ف ا د ة م E ا ل 3 م " د- ا ل " نS م ا ل 2 ا ل Sا 2 ف k ا ل \ ا م ع 9 ة ا ل 5 ي ت n ا رت o ا ب ا ل 3 2 ا ل ف ة ف 9 ه ا. ه- ا ل " نS م ا ل 2 $ ة م " $ د ة م E م 2 ا رس ة ن r ا s أ و أ ك m S م E ا لا ن r ; ة ا ل ; لا ب 9 ة ا ل 5 ي ا ر ت v 8 o ا ل 3 2 ا ل ف ة ف 9 ه ا. ا لا ن G ا ر ب $ ر ج ا ت ه ا ل m لا ث : ا لا و ل و ا ل y 2 د و ج و ا ل H ه ا ي ي. و- م m ل ي ا ل r ي ء أ و ا لا ش 9 ا ء ا ل 5 ي أ ت ل ف ه ا ا ل ; ا ل <. لا f ق ل ع E ق 9 ز- ا ل غ Sا م ة a ح- إ ل غ ا ء ا ل \ h 5 9 ل ف ي م ا د ة أ و أ ك S m م E م +ا د ا ل ف Ä ل ا ل Å G ت ق ع ف 9 ه ا ل 3 2 ا ل ف ة.. S m ا ع 8 5 ا ره ا ار س 8 ف ي م ا د ة أ و أ ك s-. S m ل د را س ي أ و أ ك Ä ف ا ل \ ا م ع ة ل 2 E م v ق ل ا ل Ñ 2 Ä ا ل ف Å- ا ل ف Ä ل ا ل H ه ا ي ي م E ا ل \ ا م ع ة ا ل # ا د ة (5 ( إ ذ ا ض ' ( ا ل * ا ل, ف ي أ ث 1 ا ء ت ا د 6 ة ا لا م : ; ا ن أ و ا لا خ : ' ا ر ف ي إ ح B A ا ل C Dا د م:ل' أو تL'K ن: N ال:;قQN انه حاول الغJ أو اش: U ك أ و ش U ع ف N ه ت D ق ع ع ل N ه ا ل ع ق \ D ا ت ا ل : ا ل N ة م : O أ- اع:'اره اس'ا ر في تلb الCادة. ب- إ ل غ ا ء ت O F ف ي H ق N ة ا ل Dا C د ا ل O F C ل ة ل ه ف ي ذ ل b ا ل ف g ل. ج- ف g ل ه م j ا ل O ا م ع ة ل C ف g ل د ا ر س ي و ا ح A ی ل ي ا ل ف g ل ا ل m l ض ' ( ف N ه. 6

7 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. 7

8 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. 8

9 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 17 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 18 9

10 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 19 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

11 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} 21 Example 1 Let A ={ 1,3,5,7]. Then 1 ϵ A, 3 ϵ A, but 2 Ï A. 11

={0, 1, 2, 3, } Z = The set of integers. = {, -3, -2, -1, 0, 1, 2, 3, } Q = The set of rational numbers.

12 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. 12

13 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 ÍA. Let S={1,2,3}, list all the subsets of S. The subsets of S are Æ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. The Empty Set The empty set is a special set. It contains no elements. It is usually denoted as { } or Æ. 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. 13

14 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 = Æ. Think of this as two events that can not happen at the same time. 14

15 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. 15

16 Examples 5 & 6 16

17 Examples 7 & 8 Examples 9 & 10 17

18 Cardinality If S is finite, then the cardinality of S, S, is the number of distinct elements in S. If S = {1,2,3} If S = {3,3,3,3,3} S = 3. S = 1. If S = Æ S = 0. If S = { Æ, {Æ}, {Æ,{Æ}} } S = 3. If S = {0,1,2,3, }, S is infinite. (more on this later) Power sets If S is a set, then the power set of S is P(S) = 2 S = { x : x Í S }. If S = {a} 2 S = {Æ, {a}}. We say, P(S) is the set of all subsets of S. If S = {a,b} If S = Æ 2 S = {Æ}. 2 S = {Æ, {a}, {b}, {a,b}}. If S = {Æ,{Æ}} 2 S = {Æ, {Æ}, {{Æ}}, {Æ,{Æ}}}. Fact: if S is finite, 2 S = 2 S. (if S = n, 2 S = 2 n ) 18

19 Example 11 19

ا ج ی ا ب ر ذ آ ی ر و ه م ج و ن ا ر ی ا ی م ال س ا ی ر و ه م ج ی د ر و م ه ع ل ا ط م ی

ن) ا ر ی ا ه- د ن س ی و ن ( ی ن س ح ال ه ش ر ز خ ه ز و ح ی ا ه ر و ش ک ی ی ا ر گ م ه ش ی ا ز ف ا و ت ا ع ز ا ن م ش ه ا ک ر د ت ا ر ک ا ذ م ر ی ث ا ت ی ی ا م ن ز ا ب ن ا ج ی ا ب ر ذ آ ی ر و ه م ج و ن ا