Discrete Structures & Algorithms

Propositional Logic

EECE 320 : UBC : Spring 2009

1

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 6 2007 is a Thursday.

•September 6 2007 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 negationsSuppose 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.”

Examplesp: 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

T F

F T

Conjunction (“and”)

p q p Λ q

F F F

F T F

T F F

T T T

Disjunction (“or”)

p q p V q

F F F

F T T

T F T

T T T

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

p q p → q

T F F

T T T

F F T

F T T

14

An equivalent for implicationIs there an expression that is equivalent to p → q but uses only the operators ¬, Λ, V?

Implication

p q p → q

F F T

F T T

T F F

T T T

p q ¬p ¬p V q

F F T T

F T T T

T F F F

T T F T

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 (p1)Statement 2 (p2)...Statement n (pn)

Statement n+1 (conclusion)

premisesor

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 (p1 Λ p2 Λ ... Λ pn → pn+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? 2n, which means that the table doubles in size with every proposition.

Implication

Two propositions are involved in an implication, therefore the truth table has 22 = 4 rows.

p qp → q

F F T

F T T

T F F

T T T

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

pq Rule of hypothetical qr syllogismpr

p q Rule of disjunctive p syllogism 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 p1, p2,…,pn 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.

Step Proved by1. sunny cold Premise #1.2. sunny Simplification of 1.3. swimsunny Premise #2.4. swim Modus tollens on 2,3.5. swimcanoe Premise #3.6. canoe Modus ponens on 4,5.7. canoeearly Premise #4.8. early Modus ponens on 6,7.

34

Proofs and theoremsA theorem is a statement that can be shown to be true. A proof is the means of doing so.

Axioms, Axioms, postulates, postulates, hypotheses hypotheses

and previously and previously proven proven

theoremstheoremsRules of Rules of

inferenceinference

ProofProof

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:

• “pq is true, and q is true, so p must be true.” (No, because FT 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:

• “pq is true, and p is false, so q must be false.” (No, again because FT 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 n2 is even.

• Attempted proof: “Assume n2 is even. Then n2=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 n2 is even.

39

Proof Methods for Implications

For proving implications pq, 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 (aq) and (bq).

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 n2 is an odd integer.

• Proof: If n is odd, then n = 2k+1 for some integer k. Thus, n2 = (2k+1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Therefore n2 is of the form 2j + 1 (with j the integer 2k2 + 2k), thus n2 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 n2 = 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 pF, which is only true if p=F

• Thus p is true.

45

Proof by Contradiction ExampleTheorem: sqrt(2) is irrational.

Proof:

Assume 21/2 were rational. This means there are integers i,j with no common divisors such that 21/2 = i /j.

Squaring both sides, 2 = i2/j2, so 2j2 = i2. So i2 is even; thus i is even.

Let i =2k. So 2j2 = (2k)2 = 4k2. Dividing both sides by 2, j2 = 2k2.

Thus j2 is even, so j is even.

But then i and j have a common divisor, namely 2, so we have a contradiction. □

2

46

Review: Proof Methods So Far

•Direct, indirect, vacuous, and trivial proofs of statements of the form pq.

•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 j3 + k3 and l3 + m3 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 pn, then p mod x = 1. So p>n, and we’re done.

51

Pancake numbers

•How did we prove the bounds on Pn?

•n Pn 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