Chapter 1 - Foundations

Transcription

1 Chapter 1 - oundations Introduction Suppose that: 1. Babies are illogical. 2. Nobody is despised who can manage a crocodile. 3. Illogical persons are despised. What conclusion may be reached using all of these premises? 1 Which is better, eternal happiness or a ham sandwich? It would appear that eternal happiness is better, but this is really not so! After all, nothing is better than eternal happiness, and a ham sandwich is certainly better than nothing. herefore a ham sandwich is better than eternal happiness. 2 And one of my favorite Lewis Carroll quotes:... Seven years and six months! Humpty Dumpty repeated thoughtfully. An uncomfortable sort of age. Now if you d asked my advice, I d have said Leave off at seven but it s too late now. I never ask advice about growing, Alice said indignantly. oo proud? the other enquired. Alice felt even more indignant at this suggestion. I mean, she said, that one can t help growing older. One can t, perhaps, said Humpty Dumpty; but two can. With proper assistance, you might have left off at seven Propositional Logic Definitions and Notation Logic Proposition Notation Negation 1 aken from Lewis Carroll 2 Raymond Smullyan 1

2 ruth ables p q p q Conjunction and Disjunction Conjunction of p and q: p q p q Disjunction of p and q: p q p q Other Definitions Exclusive Or (notation: ) Implication / Conditional (notation: or = ) Biconditional (notation: or ) p q p q p q p q 2

3 Other Operations on Implications Converse Contrapositive Inverse Precedence Operator Precedence () We will follow the book s convention and [almost] always use parentheses to group operators in order to avoid confusion. One exception to this is when negating a proposition, e.g., p q is equivalent to ( p) q, as opposed to (p q) Compound Propositions in ruth ables Example 1. p q p p q p q (p q) ( p q) 3

4 Logic and Bit Operators Applications of Propositional Logic Example 2. Suppose that: 1. Babies are illogical. 2. Nobody is despised who can manage a crocodile. 3. Illogical persons are despised. What conclusion may be reached using all of these premises? Propositional Equivalences Definitions Compound Proposition autology Contradiction Contingency 4

5 Logical Equivalence Definition 1. Notation: ruth tables can be used to show that compound propositions are logically equivalent. Example 3. Show that p p (p q): p q p q p (p q) A Larger Example... Show that (p q) (p r) p (q r): p q r p q p r (p q) (p r) q r p (q r) 5

6 able of Logical Equivalences (he abbreviations are not universal, but you may use them in your homework or on tests if you wish.) Equivalence Name Abbr. p p p p Identity / Idempotent (Conjunction) Identity / Idempotent (Disjunction) p Domination (Conjunction) DomC p Domination (Disjunction) DomD ( p) p Double Negation DN p q q p Commutative (Conjunction) CC p q q p Commutative (Disjunction) CD (p q) r p (q r) Associative (Conjunction) AC (p q) r p (q r) Associative (Disjunction) AD p (q r) (p q) (p r) Distributive (Conjunction) DC p (q r) (p q) (p r) Distributive (Disjunction) DD (p q) p q (p q) p q DeMorgan s Law (Conjunction) DeMorgan s Law (Disjunction) IdC IdD DMC DMD p (p q) p Absorption (Conjunction) AbC p (p q) p Absorption (Disjunction) AbD p p p p Negation (Conjunction) Negation (Disjunction) able 1: able of Logical Equivalences 6

7 Equivalence Name Abbr (p q) p q Negation of Implication NI p q p q Implication to Disjunction ID p q q p Contrapositive C p q p q p q (p q) (p q) (p r) p (q r) (p r) (q r) (p q) r (p q) (p r) p (q r) (p r) (q r) (p q) r able 2: Logical Equivalences Involving Implications Equivalence Name Abbr. (p q) p q Negation of Biconditional NB (p q) p q Negation of Biconditional (alternative) p q (p q) (q p) Biconditional B p q p q p q (p q) ( p q) able 3: Logical Equivalences Involving Biconditionals NB Additional autologies (Remember, tautology these will always be true for any values of p, q, r, and s.) autology Name Abbr. p p Excluded Middle EM (p q) p Simplification S p (p q) Addition A [p (p q)] q Modus Ponens MP [(p q) (q r)] (p r) Hypothetical Syllogism HS [(p q) q] p Disjunctive Syllogism DS [ q (p q)] p Modus ollens M [(p r) [(p q) (r s)]] (q s) [( q s) [(p q) (r s)]] ( p r) Constructive Dilemma Destructive Dilemma CDL DDL (p p) p Idempotent IM able 4: Additional autologies 7

8 De Morgan s Laws Example 4. Suppose there is an island populated solely by knights and knaves, and that knights always tell the truth but knaves always lie. While on this island, we encounter two people, A and B. A says I am a knave or B is a knight, while B says nothing. Determine, if possible, what A and B are. Arguments Using Logical Equivalence Example 5. Prove that (p q) q is a tautology. Example 6. Use equivalences from the tables to prove that (p q) (p r) and p (q r) are logically equivalent. 8

9 Practicality of Using ables How many rows does a truth table need for a compound proposition containing 2 variables? 3 variables? 5 variable? 100 variables? in general? Propositional Satisfiability Definition Predicates and Quantifiers Introduciton Is x > 3 a proposition? Definition 3 (Predicates (or Propositional unctions )). 9

10 Note that if x has no meaning, then P (x) is just a form. Definition 4 (Domain of Discourse). he domain of discourse (or the universe of discourse or simply domain) of x is... here are two ways to give meaning to a predicate P (x): he Universal Quantifier Definition 5. he universal quantification of the predicate P (x) is the statement... In symbols, Note: Example 7. (Let the domain of discourse be all real numbers.) 10

11 he Existential Quantifier Definition 6. he existential quantification of the predicate P (x) is the statement... In symbols, Note: Example 8. (Let the domain of discourse be all Grove City students and faculty.) Definition 7 (ree and Bound variables). Definition 8 (Scope). 11

12 Quantifiers with Conjunction and Disjunction Negating Quantified Expressions ranslating Into English Example 9. Let P (x) be the statement x likes to fly kites, Q(x, y) be the statement x knows y, and L(x, y) the statement x likes y. ranslate the following logical expressions into conversational English statements: 1. x (Q(Amy, x) P (x)) 2. x (L(Alice, x) L(x, Bob)) ranslating rom English Example 10. ranslate the following statements into logical expressions. Be sure to state the domain of discourse. 1. All cats are gray. 2. here are pigs which can fly. 12

13 1.5 - Nested Quantifiers Example x (x 0 y(xy = 1)) 2. x y(x + y = y) Note: he order of quantification matters! Example 12. Let M(x, y) = x is y s mother. ranslate the following into English: y xm(x, y) x ym(x, y) Example 13. ranslated each of the following in to English, where M is as in the previous example and S(x) = x is a student. 1. y (S(y) xm(x, y)) 2. y x (S(y) M(x, y)) 13

14 Example 14. Let L(x, y) = x loves 3 y and S as in the previous example. ranslate the following into logical expressions: 1. Everybody loves somebody. 2. here are people who love everybody. 3. All students love each other. Negating Nested Quantifiers Example 15. ( x y xy = 1) We move the negation through each level of quantification, using De Morgan s rules for quantifiers at each step: Example 16. Let I(x) = x has an internet connection, (x, y) = x and y have acebook messaged, and the domain be students in this class. ranslate the following into logical expressions: 1. Someone in your class has an internet connection but has not acebook messaged anyone else in the class. 2. here are two students in the class who, between them, have messaged everyone else in the class. 3 In a 1 John 4 sort of way. 14

15 Example 17. Let C(x, y) = student x is enrolled in class y and the domain of x be GCC students. ranslate the following into English sentences: 1. ( x y C(x, y)) 2. x y z ((x y) (C(x, z) C(y, z))) Rules of Inference Definitions Argument Conclusion Premises Valid [Argument] allacy 15

16 Standard Rules of Inference Each of the following is based on a tautology. Modus Ponens p p q q Modus ollens q p q p Hypothetical Syllogism p q q r p r Disjunctive Syllogism p q p q Addition p p q Simplification p q p Conjunction p q p q Resolution p q p r q r Example 18. Identify the rules of inference used in each of the following arguments. 1. Alice is a math major. herefore, Alice is either a math major or a c.s. major. 2. If it snows today, the college will close. he college is not closed today. herefore it did not snow today. 3. If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. herefore, if I go swimming, then I will get sunburn. 16

17 Example 19. Use rule of inference to show that the premises Henry works hard, If Henry works hard then he is a dull boy, and If Henry is a dull boy then he will not get the job imply the conclusion Henry will not get the job. Standard Rules of Inference Each of the following is based on a tautology. Universal Instantiation x P (x) P (c) for any fixed c Universal Generalization P (c) for an arbitrary c x P (x) Existential Instantiation x P (x) P (c) for some c Existential Generalization P (c) for some c x P (x) Universal Modus Ponens x (P (x) Q(x)) P (c) Q(c) Universal Modus ollens x (P (x) Q(x)) Q(c) P (c) Example 20. What can you conclude about Henry, Jack, and Jill, given the following premises? 1. Every c.s. major has an ipad. 2. Henry does not have an ipad. 3. Jill has an ipad. 4. Jack is a c.s. major. 17

18 allacies Affirming the Conclusion p q q p Denying the Hypothesis p q p q Begging the Question p p Valid or allacy? Do the following represent valid arguments, or fallacies? 1. All students in this class understand logic. Pascal is a student in this class. herefore, Pascal understands logic. (Let P (x) = x is in this class and Q(x) = x understands logic.) 2. Every c.s. major takes discrete mathematics. Esther is taking discrete mathematics. herefore, Esther is a c.s. major. (Let P (x) = x is a c.s. major and Q(x) = x takes discrete.) 3. All parrots like fruit. My pet bird is not a parrot. herefore, my pet bird does not like fruit. (Let P (x) = x is a parrot and Q(x) = x like fruit.) 4. Everyone who eats granola every day is healthy. John is not healthy. herefore John does not eat granola every day. (Let P (x) = x eats granola every day and Q(x) = x is healthy.) 18

19 1.7 - Introduction to Proofs erms that arise in ormal Proofs Proof Undefined erm heorem Proposition Lemma Corollary Conjecture Axiom / Postulate Quantifiers When no quantifier is given, then a universal quantification is assumed. Example 21. If xy > 0, then either x and y are both positive or x and y are both negative. 19

20 Basic acts/definitions/postulates An integer n is even iff there exists an integer k such that n = 2k. An integer n is odd iff there exists an integer k such that n = 2k + 1. An integer a is a perfect square iff there exists an integer b such that a = b 2. or a and b are integers such that a 0, we say that a divides b iff there exists an integer c such that b = ac. A real number r is rational iff there exists integers p and q with q 0 such that r = p q. A real number is irrational if it is not rational. Methods of Proving heorems Definition 9 (Direct Proof). o prove a statement of the form p q using a direct proof, we assume that p (the if ) is true and then show by a direct argument (which may take many steps) that q (the then ) must also be true. his is the most common form of proof, and we ll almost always start by trying this approach. Example 22. Prove the statement: If a person likes math, then he is cool. Example 23. Proposition 1. If n is a perfect square then n is either odd or divisible by 4. Proof. Q.E.D. 20

21 Methods of Proving heorems Definition 10 (Proof by Contraposition). A proof by contraposition is an indirect proof in which we prove the contrapositive of the original statement, i.e., we prove that q p (recall that the contrapositive has the same truth values as the original implication). Example 24. Prove the statement: If a person likes math, then he is cool. Example 25. Proposition 2. If n and m are integers and mn is even, then either m or n must be even. Proof. Q.E.D. 21

22 Proving a Biconditional Statement o prove a statement of the form p q we prove both p q and q p Example 26. Prove the statement: A person likes math if and only if he is cool. Proving Multiple Equivalences o prove that three or more statements are equivalent (all connected with if and only if s), it is enough to show that a chain of if... then... statements are true, as long as we can get from any statement to any other statement through the chain. Example 27. Prove that the following are equivalent [shorthand abbr. AE ; might also see AE for these are equivalent ] for any a, b R: 1. a < b a + b 2 a + b 2 > a < b 22

23 Other Methods of Proof Proof by Contradiction Assume the opposite of what you want to show and then show that this leads to a contradiction. Can be useful, but usually does not yield a very enlightening proof. As far as it is practical, try to avoid proof by contradiction. Vacuous Proof Show that p q is true by showing that p is false. rivial Proof Show that p q is true by showing that q is true (without using p). Example 28. Proposition 3. he product of a non-zero rational number and an irrational number is irrational. Proof. Q.E.D. 23

24 inding Mistakes in Proofs 1. a = b Given. 2. a 2 = ab Multiply both sides by a. 3. a 2 b 2 = ab b 2 Subtract b 2 from both sides. 4. (a b)(a + b) = b(a b) actor. 5. a + b = b Cancel a b on both sides. 6. 2b = b Substitute a for b since a = b = 1 Divide both sides by b. Where is the mistake in this proof? Other Errors in Proofs Begging the Question his occurs when the part of the proof is based on the truth of the statement being proved (we saw this in the previous section). Circular Reasoning Occurs when you use a statement to prove itself Proof Methods and Strategy Proof Methods Proof by Cases Logically, it has a form similar to: p q r p s q s r s s We can use this method when there are finitely many possibilities and show that each possible case leads to the desired result. Exhaustive Proof Similar to a proof by cases. Generally used when looking at a relatively small number of examples will exhaust all possibilities. 24

25 Something so important it gets its own subsubsection. Note Neither a Proof by Cases nor an Exhaustive Proof will constitute a valid proof unless all cases or possibilities have been examined! Proof by M&Ms (a.k.a., Proof by doing a couple of examples ) is not a valid form of proof! Example 29. heorem 1 (riangle Inequality). or any two real numbers x and y, x + y x + y. Proof. Q.E.D. Existence Proofs A proof of a proposition of the form x P (x) is called an existence proof. here are two types of existence proofs: 1. Constructive: 2. Nonconstructive: Example 30 (Constructive Example). Show that there is a positive integer that can be written as the sum of cubes in two different ways. Example 31 (Nonconstructive Example). Show that there exists two irrational numbers x and y such that x y rational. is 25

26 Uniqueness Definition 11 (he Uniqueness Quantifier).!x P (x) means Example 32. y R (y 0!x(xy = 1)) Counterexamples o show that x P (x) is false it is sufficient to find one value of x for which P (x) is false. Example 33. Every positive integer is the sum of three squares. 26

27 Open Problems Example 34 (he 3x + 1 Conjecture). Starting with any positive integer and repeatedly applying the transformation whereby an even integer gets divided by 2, and an odd integer gets multiplied by 3 and incremented by 1, we will ultimately generate the integer 1. Example 35 (Goldbach s Conjecture). Every positive even integer n 4 can be written as the sum of two prime numbers. 27

28 Additional Logic Puzzles 1. Use all of the following premises to reach a conclusion: he only books in this library, that I do not recommend for reading, are unhealthy in tone. he bound books are all well written. All the romances are healthy in tone. I do not recommend you to read any of the unbound books Use all of the following premises to reach a conclusion: All my sons are slim. No child of mine is healthy who takes no exercise. All gluttons, who are children of mine, are fat. No daughter of mine takes any exercise. 3. Let us assume that there are five houses of different colors next to each other on the same road. In each house lives a man of a different nationality. Every man has his favorite drink, his favorite brand of cigarettes, and keeps pets of a particular kind. he Englishman lives in the red house. he Swede keeps dogs. he Dane drinks tea. he green house is just to the left of the white one. he owner of the green house drinks coffee. he Pall Mall smoker keeps birds. he owner of the yellow house smokes Dunhills. he man in the center house drinks milk. he Norwegian lives in the first house. he Blend smoker has a neighbor who keeps cats. he man who smokes Blue Masters drinks bier. he man who keeps horses lives next to the Dunhill smoker. he German smokes Prince. he Norwegian lives next to the blue house. he Blend smoker has a neighbor who drinks water. Who keeps fish as his pet? 5 4 Puzzles 1 and 2 are attributed to Lewis Carroll 5 Commonly attributed to Albert Einstein. 28

29 4. he Lady or the iger. 6 A certain king likes to entertain himself by making his prisoners play a game to decide their fate. he prisoners are presented with two doors. In a room behind each door is either a lady whom the prisoner may marry, or a tiger whom may eat the prisoner. A clue is written on each door and the prisoner decides which door to open based on these clues. he clues provided to three prisoners brought before the king are below. ry to figure out which door each prisoner should open. Prisoner 1 is told that exactly one of the following clues is true and exactly one is false. Door 1: here is a lady behind this door and a tiger behind the other. Door 2: here is a lady behind one of the doors and a tiger behind the other. Prisoner 2 is told that either both clues are true or both are false. Door 1: Either there is a tiger behind this door or a lady behind the second door. Door 2: here is a lady behind this door. Prisoner 3 receives directions which are a bit tricker since the first two escaped. his prisoner is told that if a lady is behind door 1 then the clue on door 1 is true, but if a tiger is behind door 1 then the clue on that door is false. Door 2 follows the opposite rule: if a lady is behind door 2 the clue on door 2 is false, but if a tiger is behind door 2 the clue on that door is true. Door 1: A lady is waiting behind at least one of the doors. Door 2: A lady is waiting behind the other door. 6 Commonly attributed to Raymond Smullyan. 29