Logical statements and. Lecture 1 ICOM 4075

Transcription

1 Logical statements and operations Lecture 1 ICOM 4075

2 What is a statement? By a statementwe understand a sentence that affirms or denies the validity of a property or concept Examples: The color white Earth is a planet NOT A STATEMENT Barcelona is more enjoyable than Madrid

3 What is a statement? By a statementwe understand a sentence that affirms or denies the validity of a property or concept Examples: The color white Earth is a planet STATEMENT Barcelona is more enjoyable than Madrid

4 What is a statement? By a statementwe understand a sentence that affirms or denies the validity of a property or concept Examples: The color white Earth is a planet STATEMENT Barcelona is more enjoyable than Madrid

5 What is a logical statement? By a logical statement we understand a statement that can be unambiguously declared to be either true or false In the statements of the previous examples Earth is a planet LOGICAL STATEMENT Barcelona is more enjoyable than Madrid

6 What is a logical statement? By a logical statement we understand a statement that can be unambiguously declared to be either true or false In the statements of the previous examples Earth is a planet NOT A LOGICAL STATEMENT Barcelona is more enjoyable than Madrid

7 Variables Several concepts such as: Real number (a mathematical concept) Time(a physical concept) Distance( a geometrical concept) may assume (several different) values Variables are symbols (usually letters) used to represent these concepts. Consequently, variables: Have a meaning, and Assume values

8 Examples of variables Variables representing the concepts of real number, timeand distanceare illustrated below Variable Meaning Example of a value x Real number x = 5.67 t Time t= 3 hrs. 23 minutes d Distance d= 105 meters

9 What are variables good for? Variables are crucial for expressing scientific laws as mathematical relations or formulas Examples: Law Speed as a function of distance and time Exponential decay of an element Area of a circle Mathematical formula s = = d t e. 23t 2 a = xπ Variables involved d, t t x

10 Variables representing logical statements Logic studies the laws of reasoning. And as in other scientific disciplines, variablesare used in logic, to express the laws of reasoning in mathematical terms In logic, a variable represents a logical statement The logical statement that the variable represents is the meaningof the variable The value of the variableis the truth value of the statement, this is True or False Logic variables are also called Boolean variables

11 Examples Consider the following logical statements: Pluto is a planet All objects fall with the same speed -3 is greater than Below are representations with variables: Variable Meaning Value P Pluto is a planet False O All objects fall with the same speed True S -3 is greater than True

12 Basic logical operations The most basic operations of reasoning are: Negation, Conjunction, and Disjunction

13 Negation If S is a variable representing a logical statement, then the negationof S is the statement not S Examples: Statement Variable Negation Not Variable Pluto isa planet P Pluto is not a planet Not P All objects fall with the same speed -3 is greater than O Not all objects fall with the same speed Not O S -3 is not greater than Not S

14 Rules for operating with negation The next table states the changes in the values of a Boolean variable under negation S True False Not S False True

15 Example By applying these rules we get: Statement Value Negation Value P : Pluto isa planet False Not P : Pluto is not a planet True O: All objects fall with the same speed True Not O : Not all objects fall with the same speed False S: -3 is greater than True Not S : -3 is not greater than False

16 Conjunction The conjunction of two statements A and B is the statement A and B Examples: Statement A Statement B Conjunction P : Pluto isa planet O: All objects fall with the same speed O : All objects fall with the same speed S: -3 is greater than P and O : Pluto is a planet andall objects fall with the same speed O and S : All objects fall with the same speed and -3 is greater than -3.01

17 Rules for operating with conjunction The next table establishes the values of two Boolean variables after conjunction A B Aand B True True True False False True False False True False False False

18 Examples By applying these rules we get: Statement Statement Conjunction P: Pluto isa planet O: All objects fall with the same speed P and O : Pluto is a planet andall objects fall with the same speed False True False O: All objects fall with the same speed S: -3 is greater than O and S : All objects fall with the same speed and -3 is greater than True True True

19 Disjunction The disjunction of two statements A and B is the statement A or B Examples: Statement Statement Conjunction P : Pluto isa planet O : All objects fall with the same speed O: All objects fall with the same speed S: -3 is greater than P or O : Pluto is a planet orall objects fall with the same speed O or S : All objects fall with the same speed or-3 is greater than -3.01

20 Rules for operating with disjunction The next table establishes the values of two Boolean variables under disjunction A B Aor B True True True False False True False False True True True False

21 Examples By applying these rules we get: Statement Statement Conjunction P: Pluto isa planet O: All objects fall with the same speed P or O : Pluto is a planet orall objects fall with the same speed False True True O: All objects fall with the same speed S: -3 is greater than O and S : All objects fall with the same speed or -3 is greater than True True True

22 Compounded statements A compounded logical statement is one that contains at least one of the logical operations (or connectives) negation, disjunction or conjunction A basic(also called atomic) logical statement is one that has no logical operation in it Examples: A and B, A or B and Not A are all compounded statements

23 Analyzing compounded statements Consider the logical sentence: Earth is a planet or Earth is a star and there are planets that are also stars Is this statement true or false? Let s analyze (parse) the statement. This is, Identify the basic statements that compose it Identify the logical operation that link these basic statements

24 Parsing By basic statement we understand a statement that does not contains and or or s The actual parsing of the sentence is depicted below: Earth is a planet orearth is a star andthere are planets that are also stars Basic Statement 1: A : Earth is a planet Basic Statement 2: B: Earth is a star Basic Statement 3: C: there are planets that are also stars Logical operation 1: Disjunction Logical operation 2: Conjunction

25 In summary: parsing gives A table of Boolean variables Variables A B C Meaning Earth is a planet Earth is a star There are planets that are also stars and an expression of the sentence in terms of these variables: A or B and C

26 Order of precedence The order in which logic operations must be executedismade explicit by imposing parenthesis. Just as in elementary algebra, any operation within parenthesis precedes over the others A convention establishes that andhas precedence over or, unless otherwise indicated. This is an implicitprecedence, in the sense that it is assumed only if no parenthesis are imposed

27 Implicit and explicit precedence So,without parenthesis (implicit precedence), we have to EVALUATE FIRST! S = A or B and C This is the same as the parenthesized expression (explicit precedence) S= A or (B and C) Remark: (A or B) and C is not the same as S

28 So, is the sentence true? Well, if we assume the natural truth values of the basic variables A, B and C, Variable Meaning Natural value A Earth is a planet TRUE B Earth is a star FALSE C There are planets that are also stars FALSE S = A or B and C is: REMEMBER TO: Do this first! A B C B and C A or B and C True False False False True

29 Generalization But let s go beyond the particular meanings and their corresponding natural values for the variables A, B and C in the previous example Let s look at A or B and C as a general pattern of reasoningwith three Boolean variables which may occur in many other logical sentences Example: x > 1 or x < 0 and x > -.35 As a general pattern, we have to take all possible truth values for the Boolean variables in A or B and C

30 Truth values of A or B and C as a general pattern of reasoning A B C B and C A or B and C True True True True True True True False False True True False True False True True False False False True False True True True True False True False False False False False True False False False False False False False

31 Truth values of A or B and C as a general pattern of reasoning The previous example fits in this general pattern A B C B and C A or B and C True True True True True True True False False True True False True False True True False False False True False True True True True False True False False False False False True False False False False False False False

32 The notion of equivalence Two statements are said to be equivalentif they have the same truth values in all the possible choices of values for their variables Example: Not (Not A) is equivalent to A A Not A Not (Not A) True False True False True False All possible choices for one variable Same Columns = Equivalent Statement

33 Other equivalences Not (A and B) is equivalent to Not A or Not B All possible values for two variables A B A and B Not (A and B) True True True False True False False True False True False True False False False True Same Columns = Equivalent Statements A B Not A Not B Not A or Not B True True False False False True False False True True False True True False True False False True True True

34 Other equivalences (2) Not (A or B) is equivalent to Not A and Not B All possible values for two variables A B A or B Not (A or B) True True True False True False True False False True True False False False False True Same Columns = Equivalent Statement A B Not A Not B Not A and Not B True True False False False True False False True False False True True False False False False True True True

35 Summary In this lecture we have studied: The concepts of statement and logical statement The representation of statements with variables The basic operations of logic and their tables The analysis of a compounded statement (parsing) The idea of hypothesis (assumption) The evaluation of compounded statements respecting precedence The equivalence of logical sentences

36 Exercises (1) 1. Write in terms of variables and logical operations a) Real numbers are always less than or equal to the square of its value and even numbers are all divisible by 2 b) The square root of a negative number is undefined as a real number but is defined as a complex number or as an ordered pair of real numbers c) It is not the case that a planet is a star and a comet or there are planets that are stars or comets 2. Evaluate the truth tables of the previous statements

37 Exercises (2) 3. Determine whether the following pairs of statements are equivalent a) Not (A and B), Not A and Not B b) Not (A and B), Not A or Not B c) Not (A or B), Not A or Not B d) Not (A or B), Not A and Not B e) A and Not B and C, Not( Not A or B or Not C)