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Published by Penerbitan PMS, 2023-07-05 22:16:58

3_EBook Discrete Mathematics For Polytechnics

3_EBook Discrete Mathematics For Polytechnics

Keywords: mathematics

Matematik Penulis Siti Huzaifah Mohammad Nor Sharidah Mohd Roslan DISCRETE MATHEMATICS For Polytechnics


DISCRETE MATHEMATICS For Polytechnics SITI HUZAIFAH MOHAMMAD NOR SHARIDAH MOHD ROSLAN


PREFACE Assalamualaikum and peace be upon you. Alhamdulillah, finally the module of Discrete Mathematics for Polytechnics published successfully. This module is written by the lecturers from the Department of Mathematics, Science and Computer, Polytechnic Muadzam Shah and developed based on the latest Discrete Mathematics, Polytechnic Course Syllabus. The module has been developed to be one of the references to the students and for newly trained lecturers in this field. I would like to thank all the lecturers that support producing this module. Thank you for all the comment and the contribution to make this project success.


DISCRETE MATHEMATICS For Polytechnics Editor Siti Huzaifah Mohammad Writer Siti Huzaifah Mohammad Nor Sharidah Mohd Roslan Issue in 2021 All rights reserved. No part of this book may be reproduced in any form on by an electronic or mechanical means, including information storage ant retrieval systems, without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review. Published By Politeknik Muadzam Shah Lebuhraya Tun Abdul Razak 26700 Muadzam Shah Pahang Darul Makmur Tel : 09 – 450 2005/ 2006/ 2007 Fax : 09 – 450 2009


1 | P a g e DISCRETE MATHEMATI CS CONTENTS CHAPTER 1: BASIC LOGIC AND PROOF ................................................................... 2 1.1 Derive Propositional Logic ......................................................................... 2 1.2 Derive Predicate Logic ............................................................................. 11 1.3 Demonstrate Proofs ................................................................................ 16 CHAPTER 2: BOOLEAN ALGEBRA.......................................................................... 20 2.1 Carry out Boolean Functions.................................................................... 20 2.2 Construct Logic Gates.............................................................................. 23 2.3 Customize minimization of circuits.......................................................... 30 CHAPTER 3: GRAPHS AND TREES.......................................................................... 34 3.1 Derive concept of graphs......................................................................... 34 3.2 Follow concept of trees........................................................................... 48 CHAPTER 4: SETS, RELATIONS AND FUNCTIONS................................................... 68 4.1 Derive sets and set operations ................................................................ 68 4.2 Explain relations...................................................................................... 77 4.3 Carry out functions.................................................................................. 80 CHAPTER 5: BASIC COUNTING RULES................................................................... 94 5.1 Derive counting principle......................................................................... 94 5.2 Compute permutations and combinations.............................................. 98 REFERENCES....................................................................................................... 106


2 | P a g e DISCRETE MATHEMATI CS CHAPTER 1: BASIC LOGIC AND PROOF 1.1 Derive Propositional Logic Propositional Logic is the logic of compound statements built from simpler statements using Boolean connectives. Some applications in computer science: ✓ Design of digital electronic circuits. ✓ Expressing conditions in programs. ✓ Queries to databases and search engines 1.1.1 Define the purpose of proposition logic A proposition is a statement which has a value true (T) or false (F) but not both. The truth or falseness of the statement can be determined immediately. Examples of proposition: ✓ Kuala Lumpur is the capital of Malaysia – Proposition. True (T) ✓ 2016 was an Olympic year – Proposition. True (T) ✓ Two plus two equals five – Proposition. False (F) ✓ 2+6 = 5 – Proposition. False (F) The following sentences shows the example of NOT propositions: ✓ “Who’s there?” - interrogative, question ✓ “La la la la la…” - meaningless interjection ✓ “Just do it!” - imperative, command ✓ “1 + 2” - expression with a non-true/false value Propositions can be denoted using letters such as p, q, r, s etc. The truth value can be either true (T) or false (F). If a proposition is true, it is denoted by T. If the proposition is false, it is denoted by F.


3 | P a g e DISCRETE MATHEMATI CS 1.1.2 Carry out the formula in proposition logic Two propositions can be combined called compound propositions using an operators or connectives to build more complicated logical expressions. Some formulas in proposition logic are as the table below: a) Negation operator (Not: ~) • Transform a proposition into a logical negation. • It has opposite truth value from p; if p is true, p is false and vice versa. • Example: p = “I have a brown hair” ~ p = “I do not have a brown hair” • The truth table for NOT: b) Conjunction operator (And/But: ) • The conjunction operator “” (AND) combines two propositions to form their logical conjunction. • The proposition p q is true when both p and q are true and is false otherwise. • Example: p = “I will have salad for lunch.” q = “I will have steak for dinner.” p q = “I will have salad for lunch, and I will have steak for dinner.” • The truth table for AND: p ~p T F F T p q p q T T T T F F F T F F F F


4 | P a g e DISCRETE MATHEMATI CS c) Disjunction operator (Or: ) • The disjunction operator “” (OR) combines two propositions to form their logical disjunction. • The proposition p q is false when both p and q are false and is true otherwise. • Example: p = “My car has a bad engine.” q = “My car has a bad carburettor.” p q = “Either my car has a bad engine, or my car has a bad carburettor.” • The truth table for OR: d) Conditional (Implication) operator (If, then: →) • The implication p → q states that p implies q. • If p is true, then q is true; but if p is not true, then q could be either true or false. • Example: p = “You study hard.” q = “You will get a good grade.” p → q = “If you study hard, then you will get a good grade.” • p → q is false only when p is true but q is not true. • The truth table for IF, THEN : • and q is the conclusion. p q p q T T T T F T F T T F F F p q p → q T T T T F F F T T F F T


5 | P a g e DISCRETE MATHEMATI CS • From the above truth table, note that when hypothesis is true, the truth value of implication depends on the truth value of conclusion. • When hypothesis is false, the truth value of implication is always true. • Other way to express this conditional statement: “if p, then q” “p implies q” “if p, q” “p only if q” “p is sufficient for q” “q if p” “q whenever p” “q when p” “q is necessary for p” “q follows from p” “q unless ~p” e) Biconditional operator (If and only if: ) • The biconditional p ↔ q states that p is true if and only if (IFF) q is true. • Example: p = “Bush wins the 2004 election.” q = “Bush will be president for all of 2005.” p ↔ q = “If, and only if, Bush wins the 2004 election, Bush will be president for all of 2005.” p ↔ q = “Bush wins the 2004 election if and only if Bush will be president for all of 2005.” • p ↔ q means that p and q have the same truth value. • The truth table for IFF: p q p ↔ q T T T T F F F T F F F T


6 | P a g e DISCRETE MATHEMATI CS ❖ Their truth table for Boolean operations summary: ❖ Some alternative notations: ❖ Some propositions are interesting since their values in the truth table are always the same. For example as below: a) Tautologies o A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are. o A statement whose form is a tautology is called tautological statement. o Example: p ~p o Truth table: p q ~ p p q p q p → q p q T T F T T T T T F F F T F F F T T F T T F F F T F F T T NAME NOT AND OR IMPLIES IFF Propositional logic ~ → Boolean algebra p p q + C/C++/Java == Logic gates p ~p p ~p T F T F T T All TRUE so it’s a tautalogy


7 | P a g e DISCRETE MATHEMATI CS b) Contradictions o A contradiction is a compound proposition that is false no matter what! o A statement whose form is a contradiction is called contradictory statement. o Example: p ~p o Truth table: c) Contingency o A proposition that is neither a tautology nor contradiction is called a contingency. o Example: p → p o Truth table: ❖ Compound propositions that have the same truth values in all possible cases are called logically equivalent. ❖ Example: p → q is equivalent to ~q → ~p (contrapositive) p ~p p ~p T F F F T F p q p → q T T T T F F F T T F F T p q p → q ~q ~p ~q →~ p T T T F F T T F F T F F F T T F T T F F T T T T All FALSE so it’s a contradiction same values


8 | P a g e DISCRETE MATHEMATI CS Activity 1a 1. Classify whether the following sentences are a proposition or not proposition. a) Next year is 2018. b) Perak is the biggest state in Peninsular Malaysia. c) What is your name? d) Don’t miss me to much! e) Potato chip is the most delicious food in the world. 2. Let a, b and c be the propositions a: You get an A on the final examination. b: You do every exercise in this note. c: You fail the course Express each of the following proposition in English sentences. a) ~ → ~ b) ~ ↔ ( ∧ ) 3. Given p, q and r are the propositions: p: Amira will study Discrete Mathematics q: Amira will go to the beach r: Amira is in a good mood State each of the following propositions using the symbols and logical connectives. a) Amira will not go to the beach and she will study the Discrete Mathematics. b) If Amira does not study the Discrete Mathematics then Amira is not in a good mood. 4. Let p, q, and r be the following propositions: p: You get an A on the final exam q: You do every exercise in the book. r: You get an A in this class.


9 | P a g e DISCRETE MATHEMATI CS a) Write the following formulas using p, q, and r and logical connectives. i. You get an A in this class, but you do not do every exercise in the book. ii. To get an A in this class, it is necessary for you to get an A on the final. iii. Getting an A on the final and doing every exercise in the book is sufficient for getting an A in this class. b) Write in words the following: i. p → r ii. ~q → ~r iii. (~p ~q) → ~r 5. Let a, b, and c be the following propositions: a: You can have free coffee b: You are senior citizen c: It is a Tuesday Write the following proposition using a, b and c and logical connectives. “You can have free coffee if you are senior citizen and it is a Tuesday” 6. Construct a truth table for the compound propositions below then state either it is tautology, contradiction or contingency. a) A (~A B) b) (p ~q) (p → ~q) c) ~(X ~Y) X d) (~p q) (p → q) e) (A → B) ( A → C ) f) (p ↔ q) ( q ↔ r ) g) [(R S) →Q] ( Q P )


10 | P a g e DISCRETE MATHEMATI CS 7. Show that p (p → q) is a tautology and that (~q p) (p → q) is a contradiction. p q p → q p (p → q) T T T F F T F F 8. Show that p (q r) and (p q) (p r) are equivalent. 9. Find the truth tables for each of the following proposition and state whether each is a tautology, a contradiction or neither. a) A (B ¬A) b) (p → q) (~p q) c) (p ↔ q) (q → r) d) ((q p) ( q → q)) ↔ ( p q) P Q ~Q ~Q P P → Q (~Q P) (P → Q) T T T F F T F F


11 | P a g e DISCRETE MATHEMATI CS 1.2 Derive Predicate Logic 1.2.1 Describe the expression of predicate in a statement A predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables. The domain of a predicate variable is the set of all values that may be substituted in place of the variable. Predicate logic is an extension of propositional logic that permits concisely reasoning about whole classes of entities. Propositional logic treats simple propositions as atomic entities. In contrast, predicate logic distinguishes the subject of a sentence from its predicate. For example, if P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true when substituted for x. The truth set of P(x) is denoted by {x ∈ D |P(x)} which read “the set of all x in D such that P(x). 1.2.2 Identify the type of quantifier in predicate logic Quantifiers are words that refer to quantities such as some or all and tell for how many elements a given predicate is true. a) Universal • The symbol ∀ denotes “for all” and is called the universal quantifier. For example, another way to express the sentence “All human beings are mortal” is to write ∀x ∈S, x is mortal where S denotes the set of all human beings. • The domain of the predicate variable is generally indicated between the A symbol and the variable name (as in A human being x) or immediately following the variable name (as in ∀x ∈S). • For example, to find the truth and falsity of universal statements: a) Let D = {1, 2, 3, 4, 5} and consider the statement ∀x ∈ D, x2 ≥ x. Show that this statement is true. Solution: Check that “x2 ≥ x” is true for each individual x in D. 1 2 ≥ 1 2 2 ≥ 2 3 2 ≥ 3 4 2 ≥ 4 5 2 ≥ 5 Hence “∀x ∈ D, x2 ≥ x” is true.


12 | P a g e DISCRETE MATHEMATI CS b) Consider the statement ∀x ∈R, x2 ≥ x. Find a counterexample to show that this statement is false. Solution: Take x =1 2 . Then x is in R (since 1 2 is in real number) and ( 1 2 ) 2 = 1 4 ≥ 1 2 . Hence,”∀ x ∈R, x2 ≥ x” is false. b) Existential • The symbol ∃ denotes “there exists” and is called the existential quantifier. • For example, the sentence “There is a student in DBM20083 Discrete Maths” can be written as a person m such that s is a student in DBM20083 Discrete Maths or more formally ∃ m ∈ S such that s is a student in Math DBM20083 where S is the set of all people. • The domain of the predicate variable is generally indicated either between the ∃ symbol and the variable name or immediately following the variable name. • For example, to find the truth and falsity of existential statements: a) Consider the statement∃m∈ Z such that m2= m. Show that this statement is true. Solution: Observe that 1 = 1. Thus “m = m” is true for at least one integer m. Hence “∃m∈ Z such that m2= m” is true. b) Let Z = {5, 6, 7, 8, 9, 10} and consider the statement ∃m∈ Z such that m2 = m. Show that this statement is false. Solution: Note that m = m is not true for any integer’s m from 5 to 10: 5 2 = 25≠5, 62 = 36≠6, 72 = 49≠7, 8 2 = 64≠8, 92 = 81≠9, 102 = 100≠10 Thus, “∃m∈ Z such that m2= m” is false.


13 | P a g e DISCRETE MATHEMATI CS 1.2.3 Write a well-formed predicate logic in English It is important to be able to translate from formal into informal language when trying to make sense of mathematical concepts that are new to you. It is important to be able to translate from informal to formal language when thinking out a complicated problem. Rewrite the following formal statements in a variety of equivalent but more informal ways. Do not use the symbol∀and∃. Some examples are as below: a) ∀x ∈ R, x2 ≥ 0 Solution: ✓ All real numbers have nonnegative squares ✓ Every real number has a nonnegative square ✓ Any real number has a nonnegative square ✓ x has a nonnegative square, for each real number x ✓ The square of any real number is nonnegative b) ∀x ∈ R, x2 ≠-1 Solution: ✓ All real numbers have squares not equal to -1. ✓ No real numbers have squares equal to -1. ✓ The words none are to the words all are not. c) ∃m∈ Z such that m2 = m Solution: ✓ There is an integer whose square is equal to itself. ✓ We can find at least one integer equal to it. ✓ m2 =m, for some integer m. ✓ Some integer equals its own square. ✓ Some integers equal their own squares.


14 | P a g e DISCRETE MATHEMATI CS Activity 1b 1) Let P(x) denote the statement “x ≤ 4”. What are the truth values? a) P(0) b) P(4) c) P(6) 2) Let Q(x,y) denote the statement “x is the capital of y”. What are these truth values? a) b) Q(Ipoh, Perak) c) Q(Jasin, Melaka) d) Q(Kuantan, Pahang) e) Q(Kulim, Kedah) 3) Let P(x) be the statement “x spends more than five hours every weekday in class”, where the universe of discourse for x consists of all students. Express each of these quantifications in English. a) ∃ () b) ∀ () c) ∃ ¬() d) ∀ ¬() 4) Let N(x) be the statement “x has visited Langkawi”, where the universe of discourse consists of the students in your school. Express each of these quantifications in English. a) x N(x) b) x N(x) c) ~ x N(x) d) x ~N(x) e) ~ x N(x) f) x ~N(x) 5) Let P(x) be the predicate “x likes swimming” and Q(x) be the predicate “x can speak Mandarin”. The domains for both predicates are lecturers in Polytechnic. Use the quantifiers and logical connectives to express each of the following statements. a) Some lecturers in Polytechnic like swimming and can speak Mandarin. b) Every lecturer in Polytechnic like swimming if they cannot speak Mandarin.


15 | P a g e DISCRETE MATHEMATI CS 6) Let P(x), Q(x) and R(x) be the statements “x is a professor”, “x is a intelligent” and “x is a kind”, respectively. Express each of these statements using quantifier, logical connectives, and P(x), Q(x) and R(x), where the domain consists of all people. a) No professor is intelligent. b) All intelligent people are kind. c) No professor is kind. 7) Translate this specification into English where: F(p) is “Printer p is out of service” B(p) is “Printer p is busy” L(j) is “Print job j is lost” Q(j) is “Print job j is queued” a) p ( F(p) ) → j L(j) b) p B(p) → j Q(j) c) j ( Q(j) L(j) ) → p F(p) d) ( p B(p) j Q(j)) → j L(j) 8) Translate these system specifications into English where the predicate (, ) is “x is in state y” and where the universe of discourse for x and y consists of all systems and all possible states, respectively. a) ∃(, ) b) ∀((, ) ∨ (, )) c) ∃(, ) ∨ ∃(, ) d) ∃¬(, ) e) ∀¬(, )


16 | P a g e DISCRETE MATHEMATICS 1.3 Demonstrate Proofs 1.3.1 Explain the Rules of Inference to Validate Arguments An argument is a sequence of statements. To say that an argument form is valid means that no matter what particular statements are substituted for the statement variables in its premises, if the resulting premises are all true, then the conclusions is also true. Each valid logical inference rule corresponds to an implication that is a tautology. There are 6 rules of inference: a) Modus Ponens (mode of affirming) Consider the following argument form: If p then q. p. Therefore q. The fact that this argument is valid is called Modus Ponens. For example: If the last digit of this number is 0, then this number can be divided by 10. The last digit of this number is 0. This number can be divided by 10. b) Modus Tollens (Mode of Denying) If p then q. Not q. Therefore not p. For example: If Germany is located in South East Asia then Germany will join SEA Games. Germany will not join SEA Games. Germany is not located in South East Asia. premises conclusion Antecedent 1 Antecedent 2 ... consequence Inference rule p→q p q p→q ~q ~p


17 | P a g e DISCRETE MATHEMATICS c) Hypothetical Syllogism Chain of if – then statements. From the fact that one statement implies a second and the second implies a third, then we can conclude that the first statement implies the third. d) Disjunctive Syllogism/Elimination This argument says that when you have only two possibilities and you can rule one out, the other one must be the case. e) Addition If p is true, then p or q is true for any other statement q. f) Simplification If both p and q are true, then p is true. Proving Rules of Inference: → → ( → ) ∧ ( → ) → [( → ) ∧ ( → ) → ( → )] T T T T T T T T T T F T F F F T T F T F T F T T T F F F T F F T F T T T T T T T F T F T F F T T F F T T T T T T F F F T T T T T The argument is valid by Hypothetical Syllogism. p → q q → r p → r p q ~p q p q ~q p q p q p p q p q p p q q


18 | P a g e DISCRETE MATHEMATICS Activity 1c 1. What rule of inference is used in each of the following arguments? a. John likes apple pies. Therefore, John likes apple pies or ice cream. b. Mary likes chocolate and ice cream. Therefore, Mary likes chocolate. c. If it snows, then the roads are closed; it snows. Therefore, the roads are closed. d. If it snows, then the roads are closed; the roads are not closed. Therefore, it does not snow. e. To go to Hawaii, one must fly or take a boat; there is no seat on any flight to Hawaii this year. Therefore, one must take a boat to go to Hawaii this year. 2. Express the following arguments using the suitable symbols. What rules of inference are used in each of them? a) Siti have a car and house. Therefore, Siti have a car. b) If the sun is shining then Mary is at the beach; the sun is shining. Therefore, Mary is at the beach. c) If the sun is shining then Mary is at the beach. If Mary is at the beach then she is swimming. If she is swimming then she will be tired tonight. Therefore, If the sun is shining then Mary will be tired tonight. d) Either the sun is shining or raining; the sun is not shining. Therefore, it is raining. e) If it is raining, then the roads are jammed; the roads are not jammed. Therefore, it does not rain. f) Azman likes to read books. Therefore, Azman likes to read books or play football. g) If roses are red and violets are blue, then sugar is sweet and so are you; Roses are red and violets are blue. Therefore, sugar is sweet and so are you. h) If I try hard and I have talent, then I will become a musician. If I become a musician, then I will be happy. Therefore, If I will not be happy, then I did not try hard or I do not have talent. 3. Express the following arguments using the symbols indicated. What rules of inference are used? If the teens like it, then the sales volume will go up; either the teens like it or the store will close; the sales volume will not go up. Therefore, the store will close. Symbols to be used: t = The teens like it. s = The sales volume will go up. c = The store will close. 4. You are about to leave for school in the morning and discover you don’t have your glasses. You know the following statements are true: a) If my glasses are on the kitchen table, then I saw them at breakfast. b) I was reading the newspaper in the living room or I was reading the newspaper in the kitchen. c) If I was reading the newspaper in the living room then my glasses are on the coffee table. d) I did not see my glasses at breakfast.


19 | P a g e DISCRETE MATHEMATICS e) If I was reading my book in bed, then my glasses are on the bed table. f) I was reading the newspaper in the kitchen, and then my glasses are on the kitchen table. Where are the glasses? 5. Given the following argument: “You send me an email message and I will finish writing the program. If you do not send me an email message, then I will go to sleep early. Therefore, you send me an email message or I will not go to sleep early.” a) Write down all the propositions. b) Write down the premises and conclusion. c) Determine whether the argument above is valid or invalid. Scan this QR code to get the previous Final Examination Question related to this chapter


20 | P a g e DISCRETE MATHEMATICS CHAPTER 2: BOOLEAN ALGEBRA 2.1 Carry out Boolean Functions Boolean algebra provides the operations and the rules for working with the set {0, 1}. Electronic and optical switches can be studied using the set and the rules of Boolean algebra. The three operations in Boolean algebra that will use most are: • Complement of an element, denoted with a bar, is defined by: 0 − = 1 and 1 − = 0 • The Boolean sum, denoted by + or by OR, has the following values: 1 + 1 = 1,1 + 0 = 1,0 + 1 = 1,0 + 0 = 0. • The Boolean product, denoted by ⋅ or by AND, has the following values: 1 ⋅ 1 = 1, 1 ⋅ 0 = 0, 0 ⋅ 1 = 0, 0 ⋅ 0 = 0. Example 1: Find the value of 1 ⋅ 0 + (0 + 1) Solution: 1 ⋅ 0 + (0 + 1) = 0 + 1 = 0 + 0 = 0. 2.1.1 Define Boolean Expression & Boolean Functions Let B = {0,1}. Then = {(1, 2, . . . , )| ∈for1 ≤ ≤ } is the set of all possible n – tuples of 0s and 1s. The variable x is called a Boolean variable if it assumes values only from B, that is, if its only possible values are 0 and 1. A function from tois called a Boolean function of degree n. Example 2: The function (, ) = from the set of ordered pairs of Boolean variables to the set {0,1} is a Boolean function of degree 2 with(1,1) = 0, (1,0) = 1, (0,1) = 0,and (0,0) = 0. We display these values of F in table below. x y F(x,y) 1 1 0 1 0 1 0 1 0 0 0 0 • Boolean function can be represented using expressions made up from variables and Boolean operations. • The Boolean expressions in the variables 1, 2, . . . , are defined recursively as 0,1, 1, 2, . . . , are Boolean expressions; if 1and 2 are Boolean expressions, then1, (1, 2), and (1 + 2) are Boolean expressions. • Each Boolean expression represents Boolean function. The values of this function are obtained by substituting 0 and 1 for the variables in the expression.


21 | P a g e DISCRETE MATHEMATICS Example 3: Find the values of the Boolean function represented by (, , ) = + . Solution: The values of this function are displayed in Table 2. Boolean functions F and G of n variables are equal if and only if: • (1, 2, . . . , ) = (1, 2, . . . , ) whenever 1, 2, . . . , belong to B. • Two different Boolean expressions that represent the same function are called equivalent. • The Boolean expression xy, + 0 and ⋅ 1 are equivalent. • The complement of Boolean function is the function F, where (1, . . . , ) = (1, . . . , ) . • Let and be Boolean functions of degree n. The Boolean sum + and the Boolean product are defined by: i. ( + )(1, . . . , ) = (1, . . . , ) + (1, . . . , ) ii. ()(1, . . . , ) = (1, . . . , ) (1, . . . , ) 2.1.2 Use Identities of Boolean Algebra There are many identities in Boolean algebra. The most important of these are displayed in Table 3. (, , ) = + 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 0 0 0 1 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 Boolean Identities Identity Name = Law of the double complement + = ⋅ = Idempotent laws + 0 = ⋅ 1 = Identity laws + 1 = 1 ⋅ 0 = 0 Domination laws + = + = Commutative laws + ( + ) = ( + ) + () = () Associative laws + = ( + )( + ) ( + ) = + Distributive laws


22 | P a g e DISCRETE MATHEMATICS Example 4: Translate the distributive law + = ( + )( + ) in Table 3 into a logical equivalence. Solution: • Change each Boolean variable into a propositional variable. • Change the Boolean variable , and into the propositional variables , and. • Change each Boolean sum to a disjunction and each Boolean product into a conjunction. • + = ( + )( + ) ⇔ ∨ ( ∧ ) ≡ ( ∨ ) ∧ ( ∨ ) Example 5: Prove the absorption law ( + ) = using the other identities of Boolean algebra shown in Table 3. (This is called an absorption law because absorbing x + y into x leaves x unchanged). Solution: The steps used to derive this identity and the law used in each step follow: ( + ) = ( + 0)( + ) Identity law for the Boolean sum = + 0 ⋅ Distributive law of the Boolean sum over the Boolean product = + ⋅ 0 Commutative law for the Boolean product = + 0 Domination law for the Boolean product = . Identity law for the Boolean sum Activity 2a 1. Find the values of these expressions. a) 1 ⋅ 0 b) 1 + 1 c) 0 ⋅ 0 d) (1 + 0) 2. a) Show that (1 ⋅ 1) + (0.1 + 0) = 1 b) Translate the equation in part (a) into a propositional equivalence by changing each O to a F, each 1 to a T, each Boolean sum into a disjunction, each Boolean product into a conjunction, each complementation into a negation, and the equals sign to a propositional equivalence sign. 3. Use a table to express the values of each of these Boolean functions. a) (, ) = b) (, , ) = + c) (, , ) = + () d) (, , ) = ( + ) () = + ( + ) = De Morgan’s laws + = ( + ) = Absorption laws + = 1 Unit property = 0 Zero property


23 | P a g e DISCRETE MATHEMATICS 4. Use a table to show that the distributive law( + ) = + is valid. 5. Show that De Morgan’s laws hold in a Boolean algebra. That is, show that for all x and y, ( ∨ ) = ∧ and ( ∧ ) = ∨ . 2.2 Construct Logic Gates Boolean algebra is used to model the circuitry of electronic devices. Each circuit can be designed using the rules of Boolean algebra that were studied in section 2.1. The basic elements of circuits are called gates. Each type of gate implements a Boolean operation. 2.2.1 Define Logic Gates Logic gate is an electronic circuit/device which makes the logical decisions. These logic gates are digital circuits constructed from diodes, transistors, and resistors connected in such a way that the circuit output is the result of a basic logic operation (NOT, AND, OR) performed on the inputs. Each logic gates has including with the logic symbol, the corresponding truth table and the Boolean expression. A truth table is a means for describing how a logic circuit's output depends on the logic levels present at the circuit's inputs. a) Logic NOT gate • The digital Logic NOT Gate is the most basic of all the logical gates and is sometimes referred to as an Inverting Buffer or simply a Digital Inverter. • It is a single input device which has an output level that is normally at logic level "1" and goes "LOW" to a logic level "0" when its single input is at logic level "1", in other words it "inverts" (complements) its input signal. • The output from a NOT gate only returns "HIGH" again when its input is at logic level "0" giving us the Boolean expression of: Q =. • Then we can define the operation of a single input logic NOT gate as being: "If x is true, then Q is NOT true"


24 | P a g e DISCRETE MATHEMATICS a) NOT Gate Symbol Truth Table Inverter or NOT Gate Input Output x Q 1 0 0 1 Boolean Expression Q = Read as inverse of x gives Q Switch representation of NOT Gate • A is a switch which have been connected parallel with the load (LED) and source battery. When the switch is closed, current flows through the switch and no light is produce by the LED. However, when the switch is opened current flows through the LED and the light is on. b) Logic AND Gate • A Logic AND Gate is a type of digital logic gate that has an output which is normally at logic level "0" and only goes "HIGH" to a logic level "1" when ALL of its inputs are at logic level "1". • The output of a Logic AND Gate only returns "LOW" again when ANY of its inputs are at a logic level "0". The logic or Boolean expression given for a logic AND gate is that for Logical Multiplication which is denoted by a single dot or full stop symbol, (.) giving us the Boolean expression of: Q = x.y. • Then we can define the operation of a 2-input logic AND gate as being: "If both x and y are true, then Q is true" x Q =


25 | P a g e DISCRETE MATHEMATICS Symbol Truth Table 2-input AND Gate Input Output x y Q 1 1 1 1 0 0 0 1 0 0 0 0 Boolean Expression Q = x.y Read as x AND y gives Q Switch representation of AND Gate • A and B are two switches which have been connected in series with the load LED and source battery. When both the switches are closed, current flows to LED. But when one or both the switches are open, current does not flow to LED. c) Logic OR Gate • A Logic OR Gate or Inclusive-OR gate is a type of digital logic gate that has an output which is normally at logic level "0" and only goes "HIGH" to a logic level "1" when ANY of its inputs are at logic level "1". • The output of a Logic OR Gate only returns "LOW" again when ALL of its inputs are at a logic level "0". The logic or Boolean expression given for a logic OR gate is that for Logical Addition which is denoted by a plus sign, (+) giving us the Boolean expression of: Q = x + y. • Then we can define the operation of a 2-input logic OR gate as being: "If either x or y is true, then Q is true" x y Q = x.y A B


26 | P a g e DISCRETE MATHEMATICS Symbol Truth Table 2-input OR Gate Input Output x y Q 1 1 1 1 0 1 0 1 1 0 0 0 Boolean Expression Q = x + y Read as x AND y gives Q Switch representation of OR gate • A and B are two switches have been connected in parallel and this is connected in series with the load (LED) and source battery. When both the switches are open, current does not flow to LED. But, when any one of the switched is closed then current flows. d) Logic NAND Gate Symbol Truth Table 2-input NAND Gate Input Output x y Q 1 1 0 1 0 1 0 1 1 0 0 1 Boolean Expression Q = ⋅ Read as x AND y gives NOT Q x y Q = x + y x y Q = x.y A B


27 | P a g e DISCRETE MATHEMATICS e) Logic NOR Gate 2.2.2 Construct Combinations of Gates Combinational circuits can be constructed using a combination of inverters, OR gates, and AND gates. When combinations of circuits are formed, some gates may share inputs. Example 6: Construct circuits that produce the following outputs: a) ( + ) b) ( + ) c) ( + + )() Solution: a) b) c) Symbol Truth Table 2-input NOR Gate Input Output x y Q 1 1 0 1 0 0 0 1 0 0 0 1 Boolean Expression Q = + Read as x OR y gives NOT Q y x x + y x x z y z x + y + z x y x z y x y Q = x + y


28 | P a g e DISCRETE MATHEMATICS Activity 2b 1. Construct circuits from inverters, AND gates, and OR gates to produce these outputs. a) + c) + b) ( + ) d) ( + )( + ) e) ( + ) f) ( + + ) 2. Find the output of the given circuit. a) b) c) d) y x y x y x x y z A B C D


29 | P a g e DISCRETE MATHEMATICS e) f) g) h) i)


30 | P a g e DISCRETE MATHEMATICS 2.3 Customize minimization of circuits 2.3.1 Define The Minimization of Circuits The main purpose of minimizations of circuits is to produce Boolean sums of Boolean products that represent a Boolean function with the fewest products of literals such that these products contain the fewest literals possible among all sum-of-products that represent a Boolean function. Finding such a sum-of-products is called minimization of the Boolean function. Minimizing a Boolean function makes it possible to construct a circuit that uses the fewest gates and fewest inputs to the AND gates and OR gates. 2.3.2 Use The Karnaugh Map Method in Two and Three Variables The Karnaugh map or K-map is a graphical method used to simplify a Boolean expression or to convert a truth table to its corresponding logic circuit in a simple, orderly process. K-maps give us a visual method for simplifying sum-of-products expansions. The K-map, like a truth table, is a means for showing the relationship between logic inputs and desired output. Size of Map The size of the K-map with n Boolean variables is determined by 2n . The size of the group within a K-map with n Boolean variables and k number of terms in the resulting Boolean expression is determined by 2nk . Common sized maps are of 2 variables which is a 2×2 map, 3 variables which is a 2×4 map, and 4 variables which is a 4×4 map. a) K-map in Two Variables There are four possible minterms in the sum-of products expansion of a Boolean function in the two variables x and y. A K-map for a Boolean function in these two variables consists of four cells, where a 1 is placed in the cell representing a minterm if this minterm is present in the expansion. Cell are said to be adjacent if the minterms that they represent differ in exactly one literal. For instance, the cell representing is adjacent to the cells representing and. The four cells and the terms that they represent are shown in Figure 1. y x xy x x Figure 1: K-maps in Two Variables


31 | P a g e DISCRETE MATHEMATICS The next step is to identify minterms that can be combined from the K-map. Whenever there are 1s in two adjacent cells in the K-map, the minterms represented by these cells can be combined into a product involving just one of the variables. b) K-map in Three Variables A K-map in three variables is a rectangle divided into eight cells. The cells represent the eight possible minterms in three variables. Two cells are said to be adjacent if the minterms that they represent differ in exactly one literal. One of the ways to form a K-map in three variables is shown in Figure 2. To simplify a sum-of-products expansion in three variables, firstly we have to identify block of minterms that can be combined. Blocks of two adjacent cells represent pairs of minterms that can be combined into a product of two literals; 2 x 2 and 4 x 1 blocks of cells represent minterms that can be combined into a single literal; and the block of all eight cells represent a product of no literals. The goal is to identify the largest possible blocks in the map and cover all the 1s in the map with the least number of blocks, using the largest blocks first. 2.3.3 Apply The Karnaugh Map in Minimizations of The Circuits Example 7: K-map in Two variables. Find the K-map for: a) + b) + c) + + Solution: a) b) c) y 1 1 x y 1 1 x y 1 1 1 x yz x xyz x x Figure 2: K-maps in Three Variables


32 | P a g e DISCRETE MATHEMATICS Example 8: Simplify the sum-of-products expansions given in Example 7. Solution: a) b) c) Minimal expansion for these sum-of-products are: a) y b) + c) + Example 9: K-map in Three variables Use K-map to minimize these sum-of-products expansions a) + + + b) + + + + + + Solution: a) The minimal expansions into Boolean sums-of-products are: + + . b) The minimal expansions into Boolean sums-of-products are: + + . y 1 1 x y 1 1 x y 1 1 1 x yz x 1 1 1 1 x yz x 1 1 1 1 1 1 1 x


33 | P a g e DISCRETE MATHEMATICS Activity 2c 1. a) Draw a K-map for a function in two variables and put a 1 in the cell representing . a) What are the minterms represented by cells adjacent to this cell? 2. Draw the K-maps of these sum-of-products expansions in two variables. a) b) + c) + + + 3. a) Draw a K-map for a function in three variables. Put a 1 in the cell that represents . b) Which minterms are represented by cells adjacent to this cell? 4. Draw the K-maps of these sum-of-products expansions in three variables then use Kmap to minimize these sum-of-products expansions. a) + b) + c) + + + + d) + + + e) + + + f) + + + + g) + + + + +


34 | P a g e DISCRETE MATHEMATICS CHAPTER 3: GRAPHS AND TREES 3.1 Derive concept of graphs A graph can be used to represent almost any physical situation involving discrete objects and to represent a relationship among them. The following are some of the problems, among hundreds of such problems, where the graph theory can provide the solution. 1. How can we lay cable at minimum cost to make every telephone reachable from every other? 2. What is the fastest route from the national capital to each state capital? 3. How can n jobs be filled by n people with maximum total utility? 4. What is the maximum flow per unit time from source to sink in a network of pipes? 5. How many layers does a computer chip need so that wires in the same layer do not cross? 6. How can the season of a sports league be scheduled into the minimum number of weeks? 7. In what order should a travelling salesmen visit cities to minimize the travel time? 3.1.1 Define graph A graph consists of nodes called vertices or points, and edges or arcs, which connect some pairs of vertices. The vertices can be donated by a set V and edges by a set E. Thus, a graph is a set of pair of sets, represented by G = (V, E). Since each node is a point, vertices set is V= {v|v is vertex}. Similarly, each edge is represented by a pair of nodes (u,v). Thus, set of edges is a set of pairs elements (u,v) represented by, E = { (u,v)|u,v ∈ V, and there is a link between vertices u and v } F G


35 | P a g e DISCRETE MATHEMATICS 3.1.2 Identify the graph terminology a) Simple graph – A graph in which each edge connects two different vertices and where no two edges connect the same pair of vertices. b) Multigraphs – A graph that have more than one edge connecting the same pair of vertices. c) Pseudographs – A graph that have loops and multiple edges connecting the same pair of vertices. 3.1.3 Describe the properties of graph • An edge is said to be incident on each of its endpoints. • Two edges are adjacent if they are incident on the same vertex. • Two vertices are said to be adjacent if they are connected by an edge. • A vertex that is an endpoint of a loop is said to be adjacent to itself. • An isolated vertex is a vertex which is incident on no edges. • A vertex is pendant (leaf vertex) if and only if it has degree one. • the distance between two vertices in a graph is the number of edges in a shortest path connecting them. • The size of a graph is the number of its edges. • A trail is a walk without repeated edges.


36 | P a g e DISCRETE MATHEMATICS For example, given the graph, G below: It is a pair G = (V, E) where V = V(G) = set of vertices and E = E(G) = set of edges. So, V = {s, u, v, w, x, y, z} and E = {(x,s), (x,v)1, (x,v)2, (x,u), (v,w), (s,v), (s,u), (s,w), (s,y), (w,y), (u,y), (u,z),(y,z)}. There are special edges in the graphs: a) Parallel edges – Two different edges connecting the same pair of vertices. In the example, a and b are joined by two parallel edges. b) Loops – An edge that starts and ends at the same vertex. In the example, vertex d has a loop The degree of a node is the number of edges that “stick out of” the node. The sum of the degrees of all nodes in a graph is twice the number of edges of the graph (The Handshake Theorem). Let G be a graph and v a vertex of G. The degree of v, denoted deg(v) equals the number of edges that are incident on v, with an edge that is loop counted twice. The total degree of G is the sum of degrees of all the vertices of G. The Handshake Theorem: The sum of the degrees of all nodes in a graph is twice the number of edges of the graph. The degree of this vertex = 5 V1 Total degree of G = deg(v1) + deg(v2) + deg(v3) = 0 +2 + 4 = 6 V2 V3 G


37 | P a g e DISCRETE MATHEMATICS 3.1.4 Construct graph representations a) Simple graph – A graph that does not have any loops or parallel edges. b) Directed graph (Digraph) – A graph having no loops but the direction of edges is shown. c) Weighted graphs – A graph without direction of edges but there is a weight associated with each edge. For example in a road network connecting cities, the weight may be the distance or time to travel or fuel consumption during travel. d) Connected graph – A graph that is each node in the graph is connected through an edge with some other node. Simple Graph Directed Graph Connected Graph Weighted Graph Edges = 1 Total Degree = 2 Edges = 3 Total Degree = 6 Edges = 4 Total Degree = 8 Edges = 7 Total Degree = 14


38 | P a g e DISCRETE MATHEMATICS 3.1.5 Compare the different types of special graphs a) Discrete Graphs A discrete graph is a graph that only has vertices and denoted as Dn where n is the number of nodes. b) Complete Graphs A complete graph is a graph that all pairs of vertices are connected by edges (contains exactly one edge for each pair of distinct vertices) and denoted as Kn. A complete graph contains all possible edges. It does not have any loops. c) Linear Graphs A linear graph is graph that all adjacent vertices are connected by only one edge and denoted as Ln. It does not have any loops. d) Bipartite Graphs A bipartite graph is the graph which the vertex can be separated into two subsets. Each vertex in one of the subsets is connected by exactly one edge to each vertex in the other subset, but not any vertices in its own subset. It is denoted as Km,n. D3 D4 K1 K2 K3 K4 K5 L5 L3 K3,2 K3,3 L3


39 | P a g e DISCRETE MATHEMATICS 3.1.6 Explain paths, cycles and planarity in graphs Travel in a graph is accomplished by moving from one vertex to another along a sequence of adjecent edges. In the graph below, for instance, you can go from u1 to u4 by taking f1 to u2 and then f7 to u4. This is represented by writing: u1f1u2f7u4 Or you could take the roundabout route: u1f1u2f3u3f4u2f3u3f5u4f6u4f7u2f3u3f5u4 Certain types of sequences of adjacent vertices and edges are of special importance in graph theory: those that do not have a repeated edge, those that do not have a repeated vertex, and those that start and end at the same vertex. Let G be a graph, and let v and w be vertices in G. a) A walk from v to w is a finite alternating sequence of adjacent vertices and edges of G. b) A path from v to w is a walk from v to w that does not contain a repeated edge. c) A simple path from v to w is a path that does not contain a repeated vertex. d) A closed walk is a walk that starts and ends at the same vertex. e) A circuit is a closed walk that does not contain a repeated edge. f) A simple circuit is a circuit that does not have any other repeated vertex except the first and last. g) A trivial circuit is a walk consisting of a single vertex and no edge. A nontrivial circuit is a circuit with at least one edge. h) A planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect. No crossing edges. For ease of reference, these definitions are summarized in the following table: Repeated Edge? Repeated Vertex? Starts and Ends at Same Point? Walk allowed allowed allowed Path no allowed allowed Simple Path no no no Closed Walk allowed allowed yes Circuit / Cycle no allowed yes Simple Circuit no first and last only yes u1 u2 u3 u4 u5 f1 f2 f3 f4 f5 f6 f7


40 | P a g e DISCRETE MATHEMATICS Example 1: From the graph below, determine which of the following walks are paths, simple paths, circuits and simple circuits. a) v1e1v2e3v3e4v3e5v4 b) e1e3e5e5e6 c) v2v3v4v5v3v6v2 d) v2v3v4v5v6v2 e) v2v3v4v5v6v3v2 f) v1 Solution: a) This walk has repeated vertex but does not have a repeated edge, so it is a path from v1 to v4 but not a simple path. b) This is just walk from v1 to v5. It is not a path because it has repeated edge. c) This walk starts and ends at v2 and does not have a repeated edge, so it is a circuit. Since the vertex v3 is repeated in the middle, it is not a simple circuit. d) This walk starts and ends at v2, does not have a repeated edge, and does not have a repeated vertex. Thus, it is a simple circuit. e) This is just a closed walk starting and ending at v2. It is neither a circuit nor a simple circuit because edge e3 and vertex v3 are repeated. f) The first vertex of this walk is the same as its last vertex. Also, this walk has neither a repeated vertex nor a repeated edge. Thus, it is trivial circuit. 3.1.7 Ascertain the Isomorphic graphs The word isomorphism comes from the Greek meaning “same form”. Isomorphic graphs are those that have essentially the same form. Often it is required to find out whether the given two graphs are equivalent or not. They are equivalent if they have same number of edges and vertices (nodes) as well as from one graph to other there is correspondence of nodes and vertices. The mathematical definition of graph equivalence is isomorphism. The graphs are equivalent to each other if they are isomorphic to each other. Two graphs G = (V, E) and G’ = (V’, E’) are isomorphic to each other if a mapping function f from first to second graph is: i. one-to-one ii. onto iii. for each pairs of vertices u,v ∈ G, {u,v} ∈ E if and only if {f(u), f(v)} ∈ E’. This condition is called mapping of adjacency of nodes from first to second. Consider the graphs below: v1 v2 v3 v6 v4 v5 e2 e1 e3 e4 e5 e6 e7 e8 e9 e10 U2 U3 G U1 U4 G` V1 V2 V4 V3


41 | P a g e DISCRETE MATHEMATICS Let there is a mapping function f from graph G to G` with the following mappings. The above mapping is one-to-one, that is, each vertex in second maps to single vertex in first graph. Since every vertex in second graph maps to some vertex in first, it is satisfying the condition of onto. In addition, u1, u2 are adjacent and their mappings v1, v2 are adjacent: nodes u2, u3 are adjacent and their mappings v2, v3 are adjacent: and u3, u4 are adjacent and their mappings v3, v4 are adjacent. The nodes u1, u4 are adjacent and their mappings v1, v4 are adjacent. This satisfies the condition that each pair of vertices have a mapping of adjacency of nodes from first to second of isomorphism. Activity 3a 1) Identify set of vertices and set of edges for the following graphs. 2) In the following graph determine the sets of vertices, edges, loops, isolated vertices and parallel edges. a) b) U1 U2 U3 U4 V1 V2 V3 V4


42 | P a g e DISCRETE MATHEMATICS 3) Identify the degree of each vertex and the total degrees of the graphs below. 4) For each of the following graphs, determine whether or not the graph is bipartite. Explain your answers. 5) In the graph below, determine which of the following walks are walks, path, simple path, closed walk, circuit, simple circuit or trivial. a) 12462 b) a1b2c3c4d c) abcdea d) a1b2c4d6b1a e) f f) abcdeba 6) Consider the following graph. Determine which of the following walks are paths, simple paths, circuits, and/or simple circuits. a) v3v4v6v8v1v2v3. b) e1e11e14e7e8e9e10. 8 7 6 5 4 3 2 g f c b a d a) b) c) 1 d) e a) b)


43 | P a g e DISCRETE MATHEMATICS c) v1e1v2e11v8e14v6e15v2e1v1. d) v1e1v2e2v3e4v4e5v5e6v6e7v7e9v8. 7) Observe the G and G’ graphs and determine either it a isomorphic graphs or not. a) b) c)


44 | P a g e DISCRETE MATHEMATICS 3.1.8 Construct Euler paths and Euler circuits in graphs An Euler circuit for G is a circuit that contains every vertex and every edge of G. That is, an Euler circuit for G is a sequence of adjacent vertices and edges in G that starts and ends at the same vertex, uses every vertex of G at least once and uses every edge of G exactly once. If a graph has an Euler circuit, the every vertex of the graph has even degree. Both graphs below have an Euler circuit. If some vertex of a graph has odd degree, then the graph does not have an Euler circuit. An Euler path from v to w is a sequence of adjacent edges and vertices that starts at v, ends at w, passes through every vertex of G at least once and traverses every edge of G exactly once. There is an Euler path from v to w if, and only if, G is connected, v and w have odd degree, and all other vertices of G have even degree. A graph does not have an Euler circuit because v and w have odd degree. BUT there is an Euler path from v to w: vvywwvxxw v w x y Every vertex has odd degree. V1 and V3 have degree 3, which is odd. Even every vertex has even degree but this graph is Unconnected Graph V1 V2 V3 V4 c d a b 1 2 4 3 d1c2b3a4d c d a b f e 1 2 4 3 7 6 5 a1b2c3d4e5c6f7a


45 | P a g e DISCRETE MATHEMATICS 3.1.9 Construct the Hamilton paths and Hamilton circuits in graphs In such traversal, if we do not return to the starting vertex at the end, then the graph is Hamilton path. A Hamilton circuit for G is a circuit that contains every vertex of G. That is, a Hamilton circuit for G is a sequence of adjacent vertices and distinct edges in which every vertex of G appears exactly once, except for the first and the last, which are the same. Hamilton circuit for G does not need to include all the edges. Both graphs have a Hamilton circuit. For ease of reference, these definitions are summarized in the following table: PROPERTY EULER HAMILTON Repeated visits to a given node allowed? YES NO Repeated traversals of a given edge allowed? NO NO c d e a b abcdea b c d e f g h abcdefgha a b c a A graph does not have a Hamilton circuit because it does not start and end at the same vertex. BUT there is a Hamilton path because it traverses every vertex exactly once. abc


46 | P a g e DISCRETE MATHEMATICS 3.1.10 Apply graphs theories in travelling salesman problem (TSP) TSP can be modelled as a undirected weighted graph, such that cities are the graph's vertices, paths are the graph's edges, and a path's distance is the edge's length. A TSP tour becomes a Hamiltonian cycle, and the optimal TSP tour is the shortest Hamiltonian cycle. Often, the model is a complete graph where an edge connects each pair of vertices. One way to solve the general problem is to: • Write down all Hamiltonian circuits • Compute total distance for each • Pick one for which total is minimal Imagine that the graph below shows the four cities and the distance in kilometres between them. Suppose that a salesman must travel to each city exactly one, starting and ending in city A. Which route from city to city will minimize the total distance that must be travelled? This problem can be solved by writing all possible Hamilton Circuit starting and endind at A and calculate the total distance traveled for each route. ROUTE TOTAL DISTANCE (km) ABCDA 30 + 30 + 25 + 40 = 125 ABDCA 30 + 35 + 25 + 50 = 140 ACBDA 50 + 30 + 35 + 40 = 155 So, the route ABCDA is the minimum total distances of 125 km. B C A D 25 30 40 30 35 50


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