Discrete Structures & Algorithms Propositional Logic EECE 320 : UBC : Spring

2 Summary so far Pancake sorting A problem with many applications Bracketing (bounding a function) Proving bounds for pancake sorting You can make money solving such problems (Bill Gates!) Illustrated many concepts that we will learn in this course Proofs Sets Counting Performance of algorithms

3 Propositional logic Logic of compound statements built from simpler statements using Boolean connectives. Building block for mathematics and computing. Direct applications Design of digital circuits Expressing conditions in programs Database queries

4 A sandwich is better than God Nothing is better than God. A sandwich is better than nothing. Thus, a sandwich is better than God.

5 What is propositional logic? The simplest form of mathematical logic. Develops a symbolic language to treat compound and complex propositions and their logical relationship in an abstract manner. And, before we get ahead of ourselves... what is a proposition? A declarative statement that is either true or false (but not both!).

6 Examples of propositions September is a Thursday. September is a Friday. 3+2 equals 7. There is no gravity. The following are not propositions. Do your homework. What is the time? 3+4.

7 Propositions and their negations Suppose p is a proposition, then the negation of p is written as ¬p. The negation of proposition p implies that “It is not the case that p.” Examples p : It is raining. ¬p : It is not raining. p : 3+2=5. ¬p : 3+2≠5. Notice that ¬p is a proposition too.

8 Conjunctions and disjunctions Elaborate ways of saying “and” and “or” Consider two propositions, p and q Conjunction (“and”): p Λ q It is a bright and windy day. The day has to be both bright and windy. Disjunction (“or”): p V q To ride the bus you must have a ticket or hold a pass. One of the two conditions (“have a ticket” or “hold a pass”) suffices. (Though both could be true.)

9 “Exclusive or” In day-to-day speech, sometimes we use “or” as an “exclusive or”. “I will take a taxi or a bus from the airport.” Only one of “taxi” or “bus” is implied. To be precise, one would need to say “I will take either a taxi or a bus from the airport.” “Exclusive or” XOR is denoted by the symbol.

10 Truth tables Truth tables can be used to evaluate statements. A simple proposition can either be “true” or “false”. Negation p¬p TF FT Conjunction (“and”) pqp Λ q FFF FTF TFF TTT Disjunction (“or”) pqp V q FFF FTT TFT TTT Notice that the conjunction and disjunction of two propositions are also propositions (and, along with negation, are called compound propositions).

11 Logical equivalence Two statements are logically equivalent if and only if they have identical truth tables. The simplest example is ¬ ( ¬p) ≡ p.

12 Implication Another important logical construct is implication This is akin to saying “If... then...” When proposition p holds then q holds. Notation: p → q. Example: If I am on campus, I study. p : I am on campus. q : I study.

13 Truth table for implication p → q : If p, then q. If p is true, q must be true for the implication to hold. p is the assumption/premise/antecedent. q is the conclusion/consequent. Implication pqp → q TFF TTT FFT FTT

14 An equivalent for implication Is there an expression that is equivalent to p → q but uses only the operators ¬, Λ, V ? Implication pqp → q FFT FTT TFF TTT pq¬p¬p V q FFTT FTTT TFFF TTFT Consider the proposition ¬p V q

15 Proving other equivalences Easy to use truth tables and show logical equivalence. Example: distributivity p Λ (q V r) ≡ ( p V q) Λ ( p V r) Do this as an exercise. You would have seen these forms in earlier courses on digital logic design.

16 Logical equivalences The basic laws Identity Domination Idempotence Negation and double negation Commutation Association Distribution Absorption De Morgan’s laws Propositions that are logically equivalent. You will need to know them, although we will not elaborate on them in lecture. In the text (Rosen): Chapter 1, Section 2.

17 Variations on a proposition Given a proposition p → q, there are other propositions that can be stated. Example: If a function is not continuous, it is not differentiable. Contrapositive: ¬ q → ¬ p Example: If a function is differentiable, then it is continuous. Converse: q → p Example: If a function is not differentiable, then it is not continuous. Inverse: ¬ p → ¬ q Example: If a function is continuous, then it is differentiable. Of the three (contrapositive, converse, inverse), which is not like the other two? Hint: One is a logical equivalent of the original proposition. The “contrapositive” is equivalent to the original proposition.

18 Tautology & Contradiction A proposition that is always true is called a tautology. Example: p V ¬p A proposition that is always false is called a contradiction. Example: p Λ ¬p

19 Bi-conditionals (“if and only if”) If p and q are two propositions, then p ↔ q is a bi-conditional proposition. p if and only if q. ( p iff q ) p is necessary and sufficient for q. If p then q, and conversely. Example: The Thunderbirds win if and only if it is raining. p ↔ q is the same as (p → q) Λ (q → p). Are there other other equivalent statements?

20 What is an argument? A sequence of statements the ends with a conclusion. (Not the common language usage of a debate or dispute.) Structure of an argument Statement 1 ( p 1 ) Statement 2 ( p 2 )... Statement n ( p n ) Statement n+1 (conclusion) premises or antecedents

21 Example Premises “If you have a current password, you can log onto the computer network.” “You have a current password.” Conclusion “Therefore, you can log onto the computer network.”

22 Representing an argument An argument is sometimes written as follows: premises conclusion

23 Valid arguments An argument is valid if and only if it is impossible for all the premises to be true and the conclusion to be false. How do we show that an argument is valid? We can use a truth table, or We can show that (p 1 Λ p 2 Λ... Λ p n → p n+1 ) is a tautology using some rules of inference.

24 Why use rules of inference? Constructing a truth table is time consuming! If we have n propositions, what is the size of the truth table? 2 n, which means that the table doubles in size with every proposition. Two propositions are involved in an implication, therefore the truth table has 2 2 = 4 rows. Implication pq p → q FFT FTT TFF TTT

25 Rules of inference modus ponens “method of affirming”

26 Rules of inference modus tollens “method of denying”

27 Rules of inference generalization specialization

28 Rules of Inference p  qRule of hypothetical q  rsyllogism  p  r p  qRule of disjunctive  psyllogism  q More rules of inference listed in the text. Try proving them as an exercise. Chapter 1, Section 5 (Rosen).

29 Inference Rules for Quantifiers  x P(x)  P(o)(substitute any specific object o) P(g)(for g a general element.)  x P(x)  x P(x)  P(c)(substitute a new constant c) P(o) (substitute any extant object o)  x P(x) More rules of inference listed in the text. Try proving them as an exercise. Chapter 1, Section 5 (Rosen).

30 Formal Proofs A formal proof of a conclusion C, given premises p 1, p 2,…,p n consists of a sequence of steps, each of which applies some inference rule to premises or previously- proven statements to yield a new true statement (the conclusion). A proof demonstrates that if the premises are true, then the conclusion is true.

31 Formal Proof Example Suppose we have the following premises: “It is not sunny and it is cold.” “We will swim only if it is sunny.” “If we do not swim, then we will canoe.” “If we canoe, then we will be home early.” Given these premises, prove the theorem “We will be home early” using inference rules.

32 Proof Example cont. Let us adopt the following abbreviations: sunny = “It is sunny”; cold = “It is cold”; swim = “We will swim”; canoe = “We will canoe”; early = “We will be home early”. Then, the premises can be written as: (1)  sunny  cold (2) swim  sunny (3)  swim  canoe (4) canoe  early

33 Proof Example cont. StepProved by 1.  sunny  cold Premise #1. 2.  sunnySimplification of swim  sunnyPremise #2. 4.  swimModus tollens on 2,3. 5.  swim  canoe Premise #3. 6. canoeModus ponens on 4,5. 7. canoe  earlyPremise #4. 8. earlyModus ponens on 6,7.

34 Proofs and theorems A theorem is a statement that can be shown to be true. A proof is the means of doing so. Axioms, postulates, hypotheses and previously proven theorems Rules of inference Proof

35 Common Fallacies A fallacy is an inference rule or other proof method that is not logically valid. A fallacy may yield a false conclusion! Fallacy of affirming the conclusion: “p  q is true, and q is true, so p must be true.” (No, because F  T is true.) Example: If you do every problem in the book, then you will learn mathematics. You learned mathematics. Therefore you did every problem in the book (incorrect!).

36 Common Fallacies Fallacy of denying the hypothesis: “p  q is true, and p is false, so q must be false.” (No, again because F  T is true.) Example If you do every problem in the book, then you will learn mathematics. You did not do every problem in the book. Therefore you did not learn mathematics (incorrect!).

37 Common fallacies: Circular Reasoning The fallacy of (explicitly or implicitly) assuming the very statement you are trying to prove in the course of its proof. Example: Prove that an integer n is even, if n 2 is even. Attempted proof: “Assume n 2 is even. Then n 2 =2k for some integer k. Dividing both sides by n gives n = (2k)/n = 2(k/n). So there is an integer j (namely k/n) such that n=2j. Therefore n is even.” Circular reasoning is used in this proof. Where? Begs the question: How do you show that j=k/n=n/2 is an integer, without first assuming that n is even?

38 A Correct Proof Prove that an integer n is even, if n 2 is even.

39 Proof Methods for Implications For proving implications p  q, we have: Direct proof: Assume p is true, and prove q. Indirect proof: Assume  q, and prove  p. Vacuous proof: Prove  p by itself. Trivial proof: Prove q by itself. Proof by cases: Show p  (a  b), and (a  q) and (b  q).

40 Direct Proof Example Definition: An integer n is called odd iff n=2k+1 for some integer k; n is even iff n=2k for some k. Theorem: (For all numbers n) If n is an odd integer, then n 2 is an odd integer. Proof: If n is odd, then n = 2k+1 for some integer k. Thus, n 2 = (2k+1) 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k) + 1. Therefore n 2 is of the form 2j + 1 (with j the integer 2k 2 + 2k), thus n 2 is odd. □

41 Indirect Proof Example Theorem: (For all integers n) If 3n+2 is odd, then n is odd. Proof: Suppose that the conclusion is false, i.e., that n is even. Then n=2k for some integer k. Then 3n+2 = 3(2k)+2 = 6k+2 = 2(3k+1). Thus 3n+2 is even, because it equals 2j for integer j = 3k+1. So 3n+2 is not odd. We have shown that ¬(n is odd) → ¬(3n+2 is odd), thus its contra-positive (3n+2 is odd) → (n is odd) is also true. □

42 Vacuous Proof Example Theorem: (For all n) If n is both odd and even, then n 2 = n + n. Proof: The statement “n is both odd and even” is necessarily false, since no number can be both odd and even. So, the theorem is vacuously true. □

43 Trivial Proof Example Theorem: (For integers n) If n is the sum of two prime numbers, then either n is odd or n is even. Proof: Any integer n is either odd or even. So the conclusion of the implication is true regardless of the truth of the antecedent. Thus the implication is true trivially. □

44 Proof by Contradiction A method for proving p. Assume  p, and prove both q and  q for some proposition q. (Can be anything!) Why does it work? Thus  p  (q   q) (q   q) is a trivial contradiction, equal to F Thus  p  F, which is only true if  p=F Thus p is true.

45 Proof by Contradiction Example Theorem: sqrt(2) is irrational. Proof: Assume 2 1/2 were rational. This means there are integers i,j with no common divisors such that 2 1/2 = i /j. Squaring both sides, 2 = i 2 /j 2, so 2j 2 = i 2. So i 2 is even; thus i is even. Let i =2k. So 2j 2 = (2k) 2 = 4k 2. Dividing both sides by 2, j 2 = 2k 2. Thus j 2 is even, so j is even. But then i and j have a common divisor, namely 2, so we have a contradiction. □

46 Review: Proof Methods So Far Direct, indirect, vacuous, and trivial proofs of statements of the form p  q. Proof by contradiction of any statements. Next: Constructive and nonconstructive existence proofs.

47 Proving Existence A proof of a statement of the form  x P(x) is called an existence proof. If the proof demonstrates how to actually find or construct a specific element a such that P(a) is true, then it is a constructive proof. Otherwise, it is nonconstructive.

48 Constructive Existence Proof Theorem: There exists a positive integer n that is the sum of two perfect cubes in two different ways: equal to j 3 + k 3 and l 3 + m 3 where j, k, l, m are positive integers, and {j,k} ≠ {l,m} Proof: Consider n = 1729, j = 9, k = 10, l = 1, m = 12. Now just check that the equalities hold.

49 Nonconstructive Existence Proof Theorem: “There are infinitely many prime numbers.” Any finite set of numbers must contain a maximal element, so we can prove the theorem if we can just show that there is no largest prime number. I.e., show that for any prime number, there is a larger number that is also prime. More generally: For any number,  a larger prime. Formally: Show  n  p>n : p is prime.

50 The proof, using proof by cases... Given n>0, prove there is a prime p>n. Consider x = n!+1. Since x>1, we know (x is prime)  (x is composite). Case 1: x is prime. Obviously x>n, so let p=x and we’re done. Case 2: x has a prime factor p. But if p  n, then p mod x = 1. So p>n, and we’re done.

51 Pancake numbers How did we prove the bounds on P n ? n  P n  2n – 3 What are the propositions involved?

52 Wrap up Propositional logic Or propositional calculus Truth tables Logical equivalence Basic laws Rules of inference Arguments Premises and conclusions Proofs