Discrete Mathematical Structures. Chapter 1 The Foundation: Logic

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1 Discrete Mathematical Structures Chapter 1 he oundation: Logic 1

2 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction Disjunction Propositions Conditional Bio-conditional Converse Inverse Contrapositive Laws of Logic Quantifiers Universal Existential 2

3 Basic Definition: What is a Statement l A statement is a sentence having a form that is typically used to express acts l Examples: My name is Alaa I like orange 4 2 = 3 Do you live in Riyadh? 3

4 1.1 Propositional Logic Introduction l A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both. l Are the following sentences propositions? Riyadh is the capital of K.S.A. Read this carefully. (No) 1+2=3 (Yes) x+1=2 (No) What time is it? (No) (Yes) 4

5 1.1 Logical Connectives and Compound Statement l In mathematics, the letters x, y, z, are used to donate variables that can be replaced by real numbers. l hose variables can then be combined with the mathematical operations +, -, x,. l In logic, the letters p, q, r, denote the propositional variables; that can be replaced by a declarative statements. l Logical operators are used to form new propositions from two or more existing propositions. he logical operators are also called connectives. 5

6 1.1 Propositional Logic (some definitions) l Propositional Logic the area of logic that deals with propositions l Propositional Variables variables that represent propositions: p, q, r, s E.g. Proposition p oday is riday. l ruth values, 6

7 1.1 Propositional Logic (Negation) DEINIION 1 Let p be a proposition. he negation of p, denoted by p, is the statement It is not the case that p. he proposition p is read not p. he truth value of the negation of p, p is the opposite of the truth value of p. l Examples ind the negation of the proposition oday is riday. and express this in simple English. Solution: he negation is It is not the case that today is riday. In simple English, oday is not riday. or It is not riday today. ind the negation of the proposition At least 10 inches of rain fell today in Miami. and express this in simple English. Solution: he negation is It is not the case that at least 10 inches of rain fell today in Miami. In simple English, Less than 10 inches of rain fell today in Miami. 7

8 1.1 Propositional Logic (Negation) l ruth table: he ruth able for the Negation of a Proposition. p p 8

9 1.1 Propositional Logic (Conjunction) DEINIION 2 Let p and q be propositions. he conjunction of p and q, denoted by p Λ q, is the proposition p and q. he conjunction p Λ q is true when both p and q are true and is false otherwise. l Examples ind the conjunction of the propositions p and q where p is the proposition oday is riday. and q is the proposition It is raining today., and the truth value of the conjunction. Solution: he conjunction is the proposition oday is riday and it is raining today. he proposition is true on rainy ridays. 9

10 1.1 Propositional Logic (Disjunction) DEINIION 3 Let p and q be propositions. he disjunction of p and q, denoted by p ν q, is the proposition p or q. he conjunction p ν q is false when both p and q are false and is true otherwise. l Note: inclusive or : he disjunction is true when at least one of the two propositions is true. E.g. Students who have taken calculus or computer science can take this class. those who take one or both classes. exclusive or : he disjunction is true only when one of the proposition is true. E.g. Students who have taken calculus or computer science, but not both, can take this class. only those who take one of them. l Definition 3 uses inclusive or. 10

11 1.1Propositional Logic(Exclusive OR) DEINIION 4 Let p and q be propositions. he exclusive or of p and q, denoted by p q, is the proposition that is true when exactly one of p and q is true and is false otherwise. he ruth able for the Conjunction of wo Propositions. p q p Λ q he ruth able for the Disjunction of wo Propositions. p q p ν q he ruth able for the Exclusive Or (XOR) of wo Propositions. p q p q 11

12 1.1 Propositional Logic DEINIION 5 Conditional Statements Let p and q be propositions. he conditional statement p q, is the proposition if p, then q. he conditional statement is false when p is true and q is false, and true otherwise. In the conditional statement p q, p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence). i.e. A conditional statement consists of two parts, a hypothesis in the if clause and a conclusion in the then clause. or instance, If it rains, then they will cancel school. It rains, is the hypothesis. hey will cancel school, is the conclusion. l A conditional statement is also called an implication. l Example: If I am elected, then I will lower taxes. p q implication: elected, lower taxes. not elected, lower taxes. not elected, not lower taxes. elected, not lower taxes. 12

13 1.1 Propositional Logic l Example: Let p be the statement Maria learns discrete mathematics. and q the statement Maria will find a good job. Express the statement p q as a statement in English. Solution: Any of the following - If Maria learns discrete mathematics, then she will find a good job. Maria will find a good job when she learns discrete mathematics. or Maria to get a good job, it is sufficient for her to learn discrete mathematics. Maria will find a good job unless she does not learn discrete mathematics. 13

14 1.1 Propositional Logic l Other conditional statements: Converse of p q : q p o form the converse of the conditional statement, interchange the hypothesis and the conclusion. he converse of If it rains, then they will cancel school is If they cancel school, then it rains. Inverse of p q : p q o form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. he inverse of If it rains, then they will cancel school is If it does not rain, then they do not cancel school. Contrapositive of p q : q p o form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. he contrapositive of If it rains, then they will cancel school is If they do not cancel school, then it does not rain. 14

15 1.1 Propositional Logic (Biconditional Statements) DEINIION 6 Let p and q be propositions. he biconditional statement p q is the proposition p if and only if q. he biconditional statement p q is true when p and q have the same truth values, and is false otherwise. Biconditional statements are also called bi-implications. l p q has the same truth value as (p q) Λ (q p) l if and only if can be expressed by iff l Example: Let p be the statement You can take the flight and let q be the statement You buy a ticket. hen p q is the statement You can take the flight if and only if you buy a ticket. Implication: If you buy a ticket you can take the flight. If you don t buy a ticket you cannot take the flight. 15

16 1.1 Propositional Logic he ruth able for the Biconditional p q. p q p q 16

17 1.1 Propositional Logic ruth ables of Compound Propositions l We can use connectives to build up complicated compound propositions involving any number of propositional variables, then use truth tables to determine the truth value of these compound propositions. l Example: Construct the truth table of the compound proposition (p ν q) (p Λ q). he ruth able of (p ν q) (p Λ q). p q q p ν q p Λ q (p ν q) (p Λ q) 17

18 ypes of statements l A statements that is true for all possible values of its propositional variables is called tautology l (p v q) (q v p) l p v ~p l A statement that is always false is called contradiction l p ^ ~ p l A statement that can be either true or false, depending of the values of its propositional variables, is called a contingency l (p q) ^ (p ^ q) 18

19 1.1 Propositional Logic Precedence of Logical Operators l We can use parentheses to specify the order in which logical operators in a compound proposition are to be applied. l o reduce the number of parentheses, the precedence order is defined for logical operators. Precedence of Logical Operators. Operator Precedence 1 Λ ν E.g. p Λ q = ( p ) Λ q p Λ q ν r = (p Λ q ) ν r p ν q Λ r = p ν (q Λ r) 19

20 1.1 Propositional Logic ranslating English Sentences l English (and every other human language) is often ambiguous. ranslating sentences into compound statements removes the ambiguity. l Example: How can this English sentence be translated into a logical expression? You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old. Solution: Let q, r, and s represent You can ride the roller coaster, You are under 4 feet tall, and You are older than 16 years old. he sentence can be translated into: (r Λ s) q. 20

21 1.1 Propositional Logic l Example: How can this English sentence be translated into a logical expression? You can access the Internet from campus only if you are a computer science major or you are not a freshman. Solution: Let a, c, and f represent You can access the Internet from campus, You are a computer science major, and You are a freshman. he sentence can be translated into: a (c ν f). 21

22 1.1 Propositional Logic Logic and Bit Operations l Computers represent information using bits. l A bit is a symbol with two possible values, 0 and 1. l By convention, 1 represents (true) and 0 represents (false). l A variable is called a Boolean variable if its value is either true or false. l Bit operation replace true by 1 and false by 0 in logical operations. able for the Bit Operators OR, AND, and XOR. x y x ν y x Λ y x y

23 1.1 Propositional Logic DEINIION 7 A bit string is a sequence of zero or more bits. he length of this string is the number of bits in the string. l Example: ind the bitwise OR, bitwise AND, and bitwise XOR of the bit string and Solution: bitwise OR bitwise AND bitwise XOR 23

24 1.2 Propositional Equivalences DEINIION 1 Introduction A compound proposition that is always true, no matter what the truth values of the propositions that occurs in it, is called a tautology. A compound proposition that is always false is called a contradiction. A compound proposition that is neither a tautology or a contradiction is called a contingency. Examples of a autology and a Contradiction. p p p ν p p Λ p 24

25 1.2 Propositional Equivalences DEINIION 2 Logical Equivalences he compound propositions p and q are called logically equivalent if p q is a tautology. he notation p q denotes that p and q are logically equivalent. l Compound propositions that have the same truth values in all possible cases are called logically equivalent. l Example: Show that p ν q and p q are logically equivalent. ruth ables for p ν q and p q. p q p p ν q p q 25

26 1.2 Laws of Logic 1. Indempotent Laws Ø p v p p Ø p ^ p p 2. Commutative Laws p v q q v p p ^ q q ^ p 3. Association Laws (p v q ) v r p v (q v r) 26

27 1.2 Laws of Logic (Contd.) l Distributive Laws p v (q ^ r ) (p v q) ^ (p v r) p ^ (q v r) (p ^ q) v (p ^ r) l Identity Laws p v p p v P ^ p ^ p 27

28 1.2 Laws of Logic (Contd.) l Complement Laws p v ~p p ^ ~p ~ ~p p ~, ~ l De Morgan s Laws ~ (p v q) ~p ^ ~q ~ (p ^ q) ~p v ~q l p q p q (p q ) p Λ q 28

29 1.2 Propositional Equivalences Constructing New Logical Equivalences l Example: Show that (p q ) and p Λ q are logically equivalent. Solution: (p q ) ( p ν q) ( p) Λ q p Λ q by the second De Morgan law by the double negation law l Example: Show that (p Λ q) (p ν q) is a tautology. Solution: o show that this statement is a tautology, we will use logical equivalences to demonstrate that it is logically equivalent to. (p Λ q) (p ν q) (p Λ q) ν (p ν q) ( p ν q) ν (p ν q) by the first De Morgan law ( p ν p) ν ( q ν q) by the associative and communicative law for disjunction ν l Note: he above examples can also be done using truth tables. 29

30 EXAMPLES l Simplify the following statement (p v q ) ^ ~p l Show that ~p ^ (~q ^ r) v (q ^ r) v (p ^ r) r 30

31 1.3 Quantifiers l So far we have discussed the propositions in which each statement has been about a particular object. In this section we shall see how to write propositions that are about whole classes of objects. l Consider the statement: p: x is an even number he truth value of p depends on the value of x. e.g. p is true when x = 4, and false when x = 7 31

32 1.3 Universal Quantifiers l or all values of x, P(x) is true. or every x or any x x Every x l We assume that only values of x that make sense in P(x) are considered. x P(x) 32

33 1.3 Existential Quantifiers l In some situations, we only require that there is at least one value for which the predicate is true. l he existential quantification of a predicate P(x) is the statement there exists a value of x, for which P(x) is true. x P(x) here is a dog without tail ( a dog) (a dog without tail) 33

34 1.3 Negation l Negate the following: ( integers x) (x > 8) ( an integer x) (x 8) l Negate the following; ( an integer x) (0 x 8) ( integers x) (x < 0 OR x > 8) 34

35 Any Questions l Refer to chapter 1 of the book for further reading 35

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