The Logic of Compound Statements. CSE 2353 Discrete Computational Structures Spring 2018

Transcription

1 CSE 2353 Discrete Comutational Structures Sring 2018 The Logic of Comound Statements (Chater 2, E) Note: some course slides adoted from ublisher-rovided material

2 Outline 2.1 Logical Form and Logical Equivalence 2.2 Conditional Statements 2.3 Valid and Invalid Arguments Sring 2018 CSE 2353 Discrete Comutational Structures 3 Logical Form and Logical Equivalence Consider the statement: If a student is a CSE major, then the student must take CSE 2353 The logical form for this statement can be exressed as follows: Let = student is a CSE major q = student must take CSE 2353 if, then q therefore, q Sring 2018 CSE 2353 Discrete Comutational Structures 4

3 Statements In logic, a statement (sometimes called a roosition) is a sentence that is either true or false (not both) Statement: I am currently in the CSE 2353 classroom is true (if you re not skiing class) Statement: I am currently in the CSE 2353 classroom and also inside Moody Coliseum is false (unless we move the class there) Sring 2018 CSE 2353 Discrete Comutational Structures 5 Comound Statements We need a more formal (mathematical) method of exressing statements Basic comonents of a logic statement: and, or, not Symbols: and and q: q conjunction of and q or or q: q disjunction of and q not not q: q negation of q Sring 2018 CSE 2353 Discrete Comutational Structures 6

4 It is sunny and hot Examles Let = "it is sunny" and q = "it is hot" Then we can translate statement as: q It is hot or humid Let r = "it is humid". Then we have: q r It is not humid becomes r Sring 2018 CSE 2353 Discrete Comutational Structures 7 Truth Values As mentioned earlier, a statement is either true or false. We denote truth values as follows: True = T False = F We can develo a truth table to reresent truth values for comound statements. Sring 2018 CSE 2353 Discrete Comutational Structures 8

5 Negation For statement variable If is true, then is false If is false, then is true Truth table for negation: T F F T Sring 2018 CSE 2353 Discrete Comutational Structures 9 Conjunction For statement variables and q If both and q are true, then q is true Otherwise, q is false Truth table for conjunction: q q T T T T F F F T F F F F Sring 2018 CSE 2353 Discrete Comutational Structures 10

6 Disjunction For statement variables and q If at least one of and q is true, then q is true Otherwise, q is false Truth table for disjunction: q q T T T T F T F T T F F F Sring 2018 CSE 2353 Discrete Comutational Structures 11 Alication to Programming C/C++/Java Negation: if (!) {} Conjunction: if ( && q) {} Disjunction: if ( q) {} Sring 2018 CSE 2353 Discrete Comutational Structures 12

7 Evaluating the Truth of General Comound Statements Truth tables are helful for evaluating more comlex statements. A statement form (sometimes called roositional form) is an exression that contains: Statement variables (e.g.,, q, r) Logical connectives (e.g.,,, ) The truth table for a given statement form dislays the truth values for all ossible combinations of truth values for the statement variables Sring 2018 CSE 2353 Discrete Comutational Structures 13 Creating a Truth Table E.g. we have the statement form ( q) r What is the truth table? How many rows in the truth table (number of ossible combinations of truth values for, q, r)? For each variable, we have 2 ossible truth values (T, F) If we have 2 variables, we have a combination of 4 ossible truth values (TT,TF,FT,FF) General rule: for n variables, there are 2 n ossible truth value combinations Sring 2018 CSE 2353 Discrete Comutational Structures 14

8 Creating a Truth Table So for our statement form, we have 3 variables (, q, r), thus our truth table must have 2 3 = 8 rows q r ( q) r T T T T T F T F T T F F F T T F T F F F T F F F Sring 2018 CSE 2353 Discrete Comutational Structures 15 Creating a Truth Table q r q q ( q) r T T T F T T F F T F T T T F F T F T T F F T F F F F T T F F F T Sring 2018 CSE 2353 Discrete Comutational Structures 16

9 Creating a Truth Table q r q q ( q) r T T T F F T T F F F T F T T T T F F T T F T T F F F T F F F F F T T F F F F T F Sring 2018 CSE 2353 Discrete Comutational Structures 17 Creating a Truth Table q r q q ( q) r T T T F F T T T F F F F T F T T T T T F F T T T F T T F F T F T F F F F F F T T F T F F F T F F Sring 2018 CSE 2353 Discrete Comutational Structures 18

10 Examle Write the truth table for the statement form (q r) Alication to rogramming? C/C++/Java: if ( && (q r)) { } Sring 2018 CSE 2353 Discrete Comutational Structures 19 Exclusive-OR Suose that you are at a restaurant, and the menu states that your meal rice includes either tea or coffee. (If you want both, you will need to ay extra) Let = tea and q = coffee. The truth table is: q result COMMENT T T F you can t get both T F T you have tea F T T you have coffee F F F waiter forgot Sring 2018 CSE 2353 Discrete Comutational Structures 20

11 Exclusive-OR This is called Exclusive-OR, or XOR Disjunction is sometimes called Inclusive-OR XOR: can be reresented as: ( q) ( q) or ( q) ( q) Sring 2018 CSE 2353 Discrete Comutational Structures 21 Logical Equivalence Given the statement: It is cold and raining q What if we said It is raining and cold q Are we saying the same thing? Let s verify using a truth table: q q q T T T T T F F F F T F F F F F F Sring 2018 CSE 2353 Discrete Comutational Structures 22

12 Logical Equivalence q q q T T T T T F F F F T F F F F F F Note that the values of q and q are the same for each truth table row (truth values for each variable). Therefore, the statement forms q and q are said to be logically equivalent, or q q Sring 2018 CSE 2353 Discrete Comutational Structures 23 Examle Now, assume that we have two new statement forms Let X = q Y = ( q) Is X Y? Sring 2018 CSE 2353 Discrete Comutational Structures 24

13 De Morgan s Laws How do we handle negation of conjunction and disjunction? First, given variables and q, determine conjunction and disjunction of their negated values: q q q q T T F F F F T F F T F T F T T F F T F F T T T T Sring 2018 CSE 2353 Discrete Comutational Structures 25 De Morgan s Laws Next, determine the negation of the conjunction and disjunction of variables and q: q q q ( q) ( q) T T T T F F T F F T T F F T F T T F F F F F T T Sring 2018 CSE 2353 Discrete Comutational Structures 26

14 De Morgan s Laws Comare the results from the two truth tables: q q q ( q) ( q) T T F F F F T F F T T F F T F T T F F F T T T T Comaring the two tables, we notice a attern: ( q) = q and ( q) = q These roerties are called De Morgan s Laws Sring 2018 CSE 2353 Discrete Comutational Structures 27 Alication to Programming Recall our earlier examle: (q r) In C/C++/Java, this becomes if ( && (q r)) {} Now, negate using De Morgan s Laws Sring 2018 CSE 2353 Discrete Comutational Structures 28

15 Alication to Programming Negation of (q r) becomes ~[ (q r)] = ~ ~(q r) = ~ (~q ~r) In C/C++/Java, this becomes if (! (!q &&!r)) {} Sring 2018 CSE 2353 Discrete Comutational Structures 29 Examle We are given the following statement: The connector is loose or the machine is unlugged Using De Morgan s Laws, write the negation of this statement Sring 2018 CSE 2353 Discrete Comutational Structures 30

16 Tautologies and Contradictions Tautology = statement form that is always true, regardless of truth values of its statement variables Contradiction = statement form that is always false, regardless of truth values of its statement variables Sring 2018 CSE 2353 Discrete Comutational Structures 31 Tautologies and Contradictions T F T F F T T F The statement form is a tautology The statement form is a contradiction NOTATION: tautology = t contradiction = c Sring 2018 CSE 2353 Discrete Comutational Structures 32

17 Summary of Logical Equivalences Knowledge of logically equivalent statements is very useful for constructing arguments. It often haens that it is difficult to see how a conclusion follows from one form of a statement, whereas it is easy to see how it follows from a logically equivalent form of the statement. Sring CSE 2353 Discrete Comutational Structures Logical Equivalence Laws Sring 2018 CSE 2353 Discrete Comutational Structures 34 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r q r q r q r q Distributive r q r q r q r q Associative q q q q Commutative Ú Ù Ú º Ù Ú Ù Ú Ù º Ú Ù Ú Ú º Ú Ú Ù Ù º Ù Ù Ú º Ú Ù º Ù : : : Given statement variables, q, and r: Also given tautology t and contradiction c: ( ) Doublenegative Negation Identity º º Ù º Ú º Ú º Ù ~ ~ : ~ ~ : : c t c t

18 Logical Equivalence Laws A few more. Idemotent : Universal bound : De Morgan' s : ~ Ù º Ú t º t Ú º Ù c º c ( Ù q) º ~ Ú ~ q ~ ( Ú q) º ~ Ù ~ q Absortion : Ú ( Ù q) º Ù ( Ú q) Negations of t and c : ~ t º c ~ c º t º Sring 2018 CSE 2353 Discrete Comutational Structures 35 Absortion Laws ( q)? Does this make sense? Develo truth table: q q ( q) T T T T T F F T F T F F F F F F Note equivalence Sring 2018 CSE 2353 Discrete Comutational Structures 36

19 Outline 2.1 Logical Form and Logical Equivalence 2.2 Conditional Statements 2.3 Valid and Invalid Arguments Sring 2018 CSE 2353 Discrete Comutational Structures 37 Conditional Statements Recall conditional statements from earlier: general form: if, then q Denoted as q (often called imlies q ) = hyothesis q = conclusion What are the truth values for, q, q? Sring 2018 CSE 2353 Discrete Comutational Structures 38

20 Conditional Statements Use a working examle: suose the weather forecaster states This is a conditional statement: hyothesis = it is raining, conclusion q = it is flooding If it rains, then it will flood 1. First, assume both and q are true. Is the conditional statement true? It is raining, and there is a flood è the conditional statement is true Sring 2018 CSE 2353 Discrete Comutational Structures 39 Conditional Statements 2. Now, let be true and q be false. It is raining, but there is no flood è the conditional statement is false 3. Next, let and q both be false It is not raining, and there is no flood Is the conditional statement true? Yes, the forecast stated that there would be flooding if it rained it didn t make any assertion about flooding if there was no rain. Sring 2018 CSE 2353 Discrete Comutational Structures 40

21 Conditional Statements 4. Finally, let be false and q be true It is not raining, but there is a flood Is the conditional statement true? Yes, there could be another source of flooding (such as a broken water main) as before, the forecast didn t make any assertion about flooding if there was no rain. Sring 2018 CSE 2353 Discrete Comutational Structures 41 Conditional Statements Truth table becomes: q q T T T T F F F T T F F T Note that if the hyothesis () is false, then the condition ( q) is always true A conditional statement that is true by false hyothesis is called vacuously true. Sring 2018 CSE 2353 Discrete Comutational Structures 42

22 Logic Equivalence Involving Conditional Statements Can we reresent a conditional statement using our,, oerators? i.e., q? q q T T T T F F F T T F F T We note that q = F when = T and q = F, or q Thus, q = T for ( q = F), or ( q) Sring 2018 CSE 2353 Discrete Comutational Structures 43 Logic Equivalence Involving Conditional Statements Alying DeMorgan s laws, we have ( q) = ( q) = q Therefore, q q So, we can restate our earlier conditional statement to the general statement If it rains, then it will flood it is not raining or it is flooding Sring 2018 CSE 2353 Discrete Comutational Structures 44

23 Examle We are given the following statement: If n is rime, then n is odd or n is 2 Determine the negation of this statement Sring 2018 CSE 2353 Discrete Comutational Structures 45 Contraositive of a Conditional Statement The contraositive of q is q So the contraositive of our examle would be If it is not flooding, then it is not raining Is q q? Sring 2018 CSE 2353 Discrete Comutational Structures 46

24 Contraositive of a Conditional Statement q q q q T T F F T T T F F T F F F T T F T T F F T T T T Yes; a conditional statement and its contraositive are logically equivalent! Why use this? Sometimes the contraositive form is easier to use. Sring 2018 CSE 2353 Discrete Comutational Structures 47 Converse and Inverse of a Conditional Statement Converse of q is q Inverse of q is q For our examle, the converse is If it floods, then it is raining The inverse is If it is not raining, then it is not flooding Any logical equivalence? Sring 2018 CSE 2353 Discrete Comutational Structures 48

25 Converse and Inverse of a Conditional Statement q q q q q T T F F T T T T F F T F T T F T T F T F F F F T T T T T Note that a conditional statement is not logically equivalent to either its converse or inverse However, the converse and inverse are logically equivalent Sring 2018 CSE 2353 Discrete Comutational Structures 49 Only If and the Biconditional only if q means if not q, then not and if then q Mom says you will get ice cream only if you eat your lima beans = you eat your lima beans q = you get ice cream For this to be true, both and q must be true This is called the biconditional of and q: denoted as q Sring 2018 CSE 2353 Discrete Comutational Structures 50

26 Only If and the Biconditional Truth table for biconditional is: q q T T T T F F F T F F F T The biconditional form is also called if and only if or iff Sring 2018 CSE 2353 Discrete Comutational Structures 51 Necessary and Sufficient Conditions For statements and q: is a sufficient condition for q è if, then q ( q) is a necessary condition for q è if not, then not q (~ ~q) Sring 2018 CSE 2353 Discrete Comutational Structures 52

27 Necessary Condition Truth Table q ~ ~q ~ ~q q T T F F T T T F F T T T F T T F F F F F T T T T Note that ~ ~q is logically equivalent to q Thus, we can also state that is a necessary condition for q if q imlies Sring 2018 CSE 2353 Discrete Comutational Structures 53 Necessary and Sufficient Conditions Therefore, for statements and q: is a necessary condition for q è if q, then (q ) is a sufficient condition for q è if, then q ( q) Thus, for both conditions to be true: is a necessary and sufficient condition for q è iff q ( q) Sring 2018 CSE 2353 Discrete Comutational Structures 54

28 Examle: Necessary Condition Necessary condition for being a mammal: being warm-blooded Let = animal is warm-blooded, q = animal is a mammal q q T T T T F T F T F Animal is warm-blooded, but is not a mammal. Okay, since birds are also warm-blooded. F F T Animal is not warm-blooded, but is a mammal can t haen since mammals are warm-blooded by definition Sring 2018 CSE 2353 Discrete Comutational Structures 55 Examle: Sufficient Condition Sufficient condition for being a mammal: being a dog All dogs are mammals, but not all mammals are dogs. Let r = animal is dog, q = animal is a mammal r q r q T T T T F F F T T F F T Animal is dog, but is not a mammal not ossible Sring 2018 CSE 2353 Discrete Comutational Structures 56

29 Examle: Necessary and Sufficient Condition Necessary and Sufficient condition for being a mammal: having hair (fur) All mammals have hair, and any animal that has hair is a mammal Let s = animal has hair, q = animal is a mammal s q s q T T T T F F F T F F F T Sring 2018 CSE 2353 Discrete Comutational Structures 57 Outline 2.1 Logical Form and Logical Equivalence 2.2 Conditional Statements 2.3 Valid and Invalid Arguments Sring 2018 CSE 2353 Discrete Comutational Structures 58

30 Valid and Invalid Arguments Argument = a sequence of statements If yesterday was Sunday, then today is Monday Yesterday was Sunday Today is Monday First two statements are called remises (also assumtions or hyotheses) Final statement is called the conclusion Argument form = a sequence of statement forms Sring 2018 CSE 2353 Discrete Comutational Structures 59 Argument Form Let = yesterday was Sunday, q = today is Monday If then q (or q) q How do we know if an argument is valid? Develo a truth table for our argument form: Sring 2018 CSE 2353 Discrete Comutational Structures 60

31 Truth Table for Argument Premises Conclusion q q q T T T T T T F F T F F T T F T F F T F F Look at the rows for where the remises are all true these are called critical rows For every critical row, if the conclusion is true, then the argument is valid Sring 2018 CSE 2353 Discrete Comutational Structures 61 Truth Table for Argument Premises Conclusion q q q T T T T T T F F T F F T T F T F F T F F Is our argument form valid? Yes, conclusion is true for each critical row Critical row Sring 2018 CSE 2353 Discrete Comutational Structures 62

32 Another Argument Form Try another argument form: q q Truth table is: Premises Conclusion q q ~ q T T T F T T F F F F F T T T T F F T T F Sring 2018 CSE 2353 Discrete Comutational Structures 63 Valid Argument Form? Premises Conclusion q q ~ q T T T F T T F F F F F T T T T F F T T F Note that we have two critical rows. For the last critical row, the conclusion is false, so the argument form is invalid. Sring 2018 CSE 2353 Discrete Comutational Structures 64

33 Corresonding Argument? Recall: = yesterday was Sunday, q = today is Monday Argument Form q q Argument If yesterday was Sunday, then today is Monday Yesterday was not Sunday Today is Monday Sring 2018 CSE 2353 Discrete Comutational Structures 65 Examle Is the following argument valid? All Romulans have ointy ears. Sock has ointy ears. Sock is a Romulan. Sring 2018 CSE 2353 Discrete Comutational Structures 66

34 Modus Ponens The argument form that we have been using has two remises and a conclusion; this is called a syllogism. The first remise is called the major remise, and the second remise is called the minor remise. For our examle argument form: q major remise minor remise q conclusion The method that we used to check if our argument form was valid was modus onens ( method of affirming ) is the conclusion true? Sring 2018 CSE 2353 Discrete Comutational Structures 67 Modus Tollens Now, assume that we have this argument form: q q This form uses modus tollens ( method of denying ) is the conclusion false? Corresonding argument: If yesterday was Sunday, then today is Monday Today is not Monday Yesterday was not Sunday Sring 2018 CSE 2353 Discrete Comutational Structures 68

35 Rules of Inference Rule of inference = a form of argument that is valid Modus onens and modus tollens are both rules of inference There are also other rules of inference Sring 2018 CSE 2353 Discrete Comutational Structures 69 Rules of Inference Other Tyes 1. Generalization 2. Secialization 3. Elimination 4. Transitivity 5. Proof by Division into Cases Sring 2018 CSE 2353 Discrete Comutational Structures 70

36 Generalization q let = student is a CSE major q = student is an EE major remise = student is a CSE major conclusion (generalization) = student is a CSE major or EE major Sring 2018 CSE 2353 Discrete Comutational Structures 71 q q Secialization Student is both a CSE major and an EE major student is an EE major Sring 2018 CSE 2353 Discrete Comutational Structures 72

37 Elimination q q Let = student is a junior, q = student is a senior Student is a junior or a senior Student is not a senior student is a junior Sring 2018 CSE 2353 Discrete Comutational Structures 73 q q r r Transitivity Let = it rains, q = it floods, r = Riverside Ave. is closed If it rains, then it floods If it floods, then Riverside Ave. is closed If it rains, then Riverside Ave. is closed Sring 2018 CSE 2353 Discrete Comutational Structures 74

38 q r q r r Proof by Division into Cases Let = today is a university holiday, q = today is a university break, r = there are no classes today Today is a university holiday or break If today is a university holiday, then there are no classes today If today is a university break, then there are no classes today There are no classes today Sring 2018 CSE 2353 Discrete Comutational Structures 75 Fallacies Fallacy = error in reasoning that results in an invalid argument Tyes: 1. Converse Error 2. Inverse Error Sring 2018 CSE 2353 Discrete Comutational Structures 76

39 If it rains, then it floods There is a flood It is raining Converse Error It could flood without rain, such as a water main break Let = it rains, q = it floods q q Sring 2018 CSE 2353 Discrete Comutational Structures 77 Converse Error Truth Table Premises Conclusion q q q T T T T T T F F F T F T T T F F F T F F Note critical row where conclusion is false, so this argument form is invalid Sring 2018 CSE 2353 Discrete Comutational Structures 78

40 If it rains, then it floods It is not raining There is no flood Inverse Error As before, it could flood without rain, such as a water main break Let = it rains, q = it floods q q Sring 2018 CSE 2353 Discrete Comutational Structures 79 Inverse Error Truth Table Premises Conclusion q q ~ ~q T T T F F T F F F T F T T T F F F T T T Note critical row where conclusion is false, so this argument form is invalid Sring 2018 CSE 2353 Discrete Comutational Structures 80