MATH 213 - Introduction, Chapter 1, and Additional Logic ... · Suppose there is an island...

Chapter 1 - Foundations

MATH 213Introduction, Chapter 1, and Additional Logic Material

Dr. Eric Bancroft

Fall 2013

Dr. Eric Bancroft MATH 213 Fall 2013 1 / 88

Chapter 1 - Foundations

Chapter 1 - FoundationsIntroduction1.1 - Propositional Logic1.2 - Applications of Propositional Logic1.3 - Propositional Equivalences1.4 - Predicates and Quantifiers1.5 - Nested Quantifiers1.6 - Rules of Inference1.7 - Introduction to Proofs1.8 - Proof Methods and Strategy


Chapter 1 - Foundations 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? Itwould appear that eternal happiness is better, but this is reallynot so! After all, nothing is better than eternal happiness, anda ham sandwich is certainly better than nothing. Therefore aham sandwich is better than eternal happiness.2

1Taken from Lewis Carroll2Raymond Smullyan


Chapter 1 - Foundations Introduction

And one of my favorite Lewis Carroll quotes:

. . . “Seven years and six months!” Humpty Dumpty repeatedthoughtfully. “An uncomfortable sort of age. Now if you’dasked my advice, I’d have said ‘Leave off at seven’ but it’s toolate now.”“I never ask advice about growing,” Alice said indignantly.“Too 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.”


Chapter 1 - Foundations 1.1 - Propositional Logic

Definitions and Notation

I Logic

I Proposition

I Notation

I Negation



Truth Tables

p q ¬p ¬q



Conjunction and Disjunction

Conjunction of p and q:

p q p ∧ q

T T

T F

F T

F F

Disjunction of p and q:

p q p ∨ q

T T

T F

F T

F F



Other Definitions

I Exclusive Or (notation: ⊕)

I Implication / Conditional (notation: → or =⇒ )

I Biconditional (notation: ↔ or ⇐⇒ )



Example

p q p⊕ q p→ q p↔ q

T T

T F

F T

F F



Other Operations on Implications

I Converse

I Contrapositive

I Inverse



Precedence

Operator Precedence

() 0

¬ 1

∧ 2

∨ 3

→ 4

↔ 5

We will follow the book’s convention and [almost] always useparentheses to group operators in order to avoid confusion. Oneexception to this is when negating a proposition, e.g., ¬p ∧ q isequivalent to (¬p) ∧ q, as opposed to ¬(p ∧ q)



Compound Propositions in Truth Tables

Example

p q ¬p p→ q ¬p ∧ q (p→ q) ∨ (¬p ∧ q)

T T

T F

F T

F F



Logic and Bit Operators


Chapter 1 - Foundations 1.2 - Applications of Propositional Logic

Example

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?


Chapter 1 - Foundations 1.3 - Propositional Equivalences

Definitions

I Compound Proposition

I Tautology

I Contradiction

I Contingency



Logical Equivalence

Definition

Notation: ≡



Truth tables can be used to show that compound propositions arelogically equivalent.

Example

Show that p ≡ p ∨ (p ∧ q):

p q p ∧ q p ∨ (p ∧ q)

T T

T F

F T

F F



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)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F



Table of Logical Equivalences I

(The abbreviations are not universal, but you may use them in yourhomework or on tests if you wish.)

Equivalence Name Abbr.

p ∧ T ≡ p Identity / Idempotent(Conjunction)

IdC

p ∨ F ≡ p Identity / Idempotent(Disjunction)

IdD

p ∧ F ≡ F Domination (Conjunction) DomC

p ∨ T ≡ T Domination (Disjunction) DomD

¬(¬p) ≡ p Double Negation DN

p ∧ q ≡ q ∧ p Commutative (Conjunction) CC



Table of Logical Equivalences II

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 DeMorgan’s Law(Conjunction)

DMC

¬(p ∨ q) ≡ ¬p ∧ ¬q DeMorgan’s Law(Disjunction)

DMD

p ∧ (p ∨ q) ≡ p Absorption (Conjunction) AbC

p ∨ (p ∧ q) ≡ p Absorption (Disjunction) AbD



Table of Logical Equivalences III

p ∧ ¬p ≡ F Negation (Conjunction)

p ∨ ¬p ≡ T Negation (Disjunction)



Logical Equivalences Involving Implications

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



Logical Equivalences Involving Biconditionals

Equivalence Name Abbr.

¬(p↔ q) ≡ ¬p↔ q Negation of Biconditional NB

¬(p↔ q) ≡ p↔ ¬q Negation of Biconditional(alternative)

NB

p↔ q ≡ (p→ q) ∧ (q → p) Biconditional B

p↔ q ≡ ¬p↔ ¬qp↔ q ≡ (p ∧ q) ∨ (¬p ∧ ¬q)



Additional Tautologies I

(Remember, ‘tautology’ these will always be true for any values of p, q,r, and s.)

Tautology 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 Tollens MT



Additional Tautologies II

[(p ∨ r) ∧ [(p→ q) ∧ (r → s)]]→ (q ∨ s)

Constructive Dilemma CDL

[(¬q ∨ ¬s) ∧ [(p→ q) ∧ (r → s)]]→ (¬p ∨ ¬r)

Destructive Dilemma DDL

(p ∨ p)→ p Idempotent IM



De Morgan’s Laws

Example

Suppose there is an island populated solely by knights and knaves, andthat knights always tell the truth but knaves always lie. While on thisisland, we encounter two people, A and B. A says “I am a knave or Bis a knight”, while B says nothing. Determine, if possible, what A andB are.



Arguments Using Logical Equivalence

Example

Prove that ¬(p→ q)→ ¬q is a tautology.



Example

Use equivalences from the tables to prove that (p→ q) ∧ (p→ r) andp→ (q ∧ r) are logically equivalent.



Practicality of Using Tables

How many rows does a truth table need for a compound propositioncontaining 2 variables? 3 variables? 5 variable? 100 variables? ingeneral?



Propositional Satisfiability

Definition


Chapter 1 - Foundations 1.4 - Predicates and Quantifiers

Introduciton

Is “x > 3” a proposition?

Predicates (or ‘Propositional Functions’)



Note that if x has no meaning, then P (x) is just a form.

DefinitionThe domain of discourse (or the universe of discourse or simplydomain) of x is . . .

There are two ways to give meaning to a predicate P (x):

1.

2.



The Universal Quantifier

DefinitionThe universal quantification of the predicate P (x) is the statement . . .

In symbols,

Note



Example

(Let the domain of discourse be all real numbers.)



The Existential Quantifier

DefinitionThe existential quantification of the predicate P (x) is the statement . . .

In symbols,

Note



Example

(Let the domain of discourse be all Grove City students.)



Definition (Free and Bound variables)

Definition (Scope)



Quantifiers with Conjunction and Disjunction



Negating Quantified Expressions



Translating Into English

Example

Let P (x) be the statement “x likes to fly kites”, Q(x, y) be thestatement “x knows y”, and L(x, y) the statement “x likes y”.Translate the following logical expressions into English statements:

1. ∀x (Q(Joan, x)→ P (x))

2. ∀x (L(Susie, x)→ ¬L(x,Calvin))



Translating From English

Example

Translate the following statements into logical expressions

1. “All cats are gray.”

2. “There are pigs which can fly.”


Chapter 1 - Foundations 1.5 - Nested Quantifiers

Example

1. ∀x (x 6= 0→ ∃y(xy = 1))

2. ∀x∀y(x + y = y)

Note: The order of quantification matters!

Example

Let M(x, y) = “x is y’s mother”. Translate the following into English:

I ∀y∃xM(x, y)

I ∃x∀yM(x, y)



Example

Translated each of the following in to English, where M is as in theprevious example and S(x) = “x is a student”.

1. ∀y (S(y)→ ∃xM(x, y))

2. ∀y∃x (S(y) ∧M(x, y))



[Clip]



Example

Let L(x, y) = “x loves3 y” and S as in the previous example. Translatethe following into logical expressions:

1. Everybody loves somebody.

2. There are people who love everybody.

3. All students love each other.

3In a 1 John 4 sort of way.Dr. Eric Bancroft MATH 213 Fall 2013 44 / 88


Negating Nested Quantifiers

Example

¬(∀x∃y xy = 1)We move the negation through each level of quantification, using DeMorgan’s rules for quantifiers at each step:



Example

Let I(x) = “x has an internet connection”,F (x, y) = “x and y have Facebook messaged”, and the domain bestudents in this class. Translate the following into logical expressions:

1. Someone in your class has an internet connection but has notFacebook messaged anyone else in the class.

2. There are two students in the class who, between them, havemessaged everyone else in the class.



Example

Let C(x, y) = “student x is enrolled in class y” and the domain beGCC students. Translate the following into English sentences:

1. ¬ (∃x∀y C(x, y))

2. ∃x∃y∀z ((x 6= y) ∧ (C(x, z)↔ C(y, z)))


Chapter 1 - Foundations 1.6 - Rules of Inference

Definitions

I Argument

I Premises

I Conclusion

I Valid [Argument]

I Fallacy



Standard Rules of Inference I

Each of the following is based on a tautology.

I Modus Ponenspp→ q

∴ q

I Modus Tollens¬qp→ q

∴ ¬p

I Hypothetical Syllogismp→ qq → r

∴ p→ r



Standard Rules of Inference II

I Disjunctive Syllogismp ∨ q¬p

∴ q

I Additionp

∴ p ∨ q

I Simplificationp ∧ q

∴ p



Standard Rules of Inference III

I Conjunctionpq

∴ p ∧ q

I Resolutionp ∨ q¬p ∨ r

∴ q ∨ r



Examples

Identify the rules of inference used in each of the following arguments.

1. Alice is a math major. Therefore, Alice is either a math major ora c.s. major.

2. If it snows today, the college will close. The college is not closedtoday. Therefore it did not snow today.

3. If I go swimming, then I will stay in the sun too long. If I stay inthe sun too long, then I will sunburn. Therefore, if I go swimming,then I will get sunburn.



Example

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 dullboy then he will not get the job” imply the conclusion “Henry will notget the job.”



Standard Rules of Inference I

Each of the following is based on a tautology.

I Universal Instantiation∀xP (x)

∴ P (c) for any fixed c

I Universal GeneralizationP (c) for an arbitrary c

∴ ∀xP (x)

I Existential Instantiation∃xP (x)

∴ P (c) for some c



Standard Rules of Inference II

I Existential GeneralizationP (c) for some c

∴ ∃xP (x)

I Universal Modus Ponens∀x (P (x)→ Q(x))P (c)

∴ Q(c)

I Universal Modus Tollens∀x (P (x)→ Q(x))¬Q(c)

∴ ¬P (c)



Example

What can you conclude about Henry, Jack, and Jill, given the followingpremises?

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.



Fallacies

I Affirming the Conclusionp→ qq

∴ p

I Denying the Hypothesisp→ q¬p

∴ ¬q

I Begging the Questionp

∴ p



Valid or Fallacy? I

Do the following represent valid arguments, or fallacies?

1. All students in this class understand logic. Pascal is a student inthis class. Therefore, Pascal understands logic. (LetP (x) = “x is in this class” and Q(x) = “x understands logic”.)

2. Every c.s. major takes discrete mathematics. Esther is takingdiscrete mathematics. Therefore, Esther is a c.s. major. (LetP (x) = “x is a c.s. major” and Q(x) = “x takes discrete”.)



Valid or Fallacy? II

3. All parrots like fruit. My pet bird is not a parrot. Therefore, mypet bird does not like fruit. (Let P (x) = “x is a parrot” andQ(x) = “x like fruit”.)

4. Everyone who eats granola every day is healthy. John is nothealthy. Therefore John does not eat granola every day. (LetP (x) = “x eats granola every day” and Q(x) = “x is healthy”.)



Example

Let

I F (x) =“x’s cable freezes”I I(x) =“x is irritable”I S(x) =“x’s work suffers”I W (x) =“x is wrongly

convicted”

I T (x, y) =“x has time to thinkabout y (a lot)”

I E(x) =“x’s house explodes”

I D(x) =“x has DirecTV R©”


Chapter 1 - Foundations 1.7 - Introduction to Proofs

Terms that arise in Formal Proofs I

I Proof

I Undefined Term

I Theorem

I Proposition



Terms that arise in Formal Proofs II

I Lemma

I Corollary

I Conjecture

I Axiom / Postulate



Quantifiers

When no quantifier is given, then a universal quantification is assumed.

Example

If xy > 0, then either x and y are both positive or x and y are bothnegative.



Basic Facts/Definitions/Postulates

I An integer n is even if there exists an integer k such that n = 2k.

I An integer n is odd if there exists an integer k such that n = 2k+ 1.

I An integer a is a perfect square if there exists an integer b suchthat a = b2.

I If a and b are integers such that a 6= 0, we say that a divides b ifthere exists an integer c such that b = ac.

I A real number r is rational if there exists integers p and q withq 6= 0 such that r = p

q . A real number is irrational if it is notrational.



Methods of Proving Theorems

Direct ProofTo prove a statement of the form p→ q using a direct proof, weassume 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.This is the most common form of proof, and we’ll almost always startby trying this approach.

Example

Prove the statement: “If a person likes math, then he is cool.”



Example

Proposition

If n is a perfect square then n is either odd or divisible by 4.

Proof.

Q.E.D.



Methods of Proving Theorems

Proof by Contraposition

A proof by contraposition is an indirect proof in which we prove thecontrapositive of the original statement, i.e., we prove that ¬q → ¬p(recall that the contrapositive has the same truth values as the originalimplication).

Example

Prove the statement: “If a person likes math, then he is cool.”



Example

Proposition

If n and m are integers and mn is even, then either m or n must beeven.

Proof.

Q.E.D.



Proving a Biconditional Statement

To prove a statement of the form p↔ q we prove both

I p→ q and

I q → p

Example

Prove the statement: “A person likes math if and only if he is cool.”



Proving Multiple Equivalences

To prove that three or more statements are equivalent (all connectedwith “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 anystatement to any other statement through the chain.



Example

Prove that the following are equivalent [shorthand abbr. “TFAE”;might also see “TAE” for “these are equivalent”] for any a, b ∈ R:

1. a < b

2.a + b

2> a

3.a + b

2< b



Other Methods of Proof

Proof by Contradiction

Assume the opposite of what you want to show and then show thatthis leads to a contradiction. Can be useful, but usually does not yielda very ‘enlightening’ proof. As far as it is practical, try to avoid proofby contradiction.

Vacuous ProofShow that p→ q is true by showing that p is false.

Trivial ProofShow that p→ q is true by showing that q is true (without using p).



Example

Proposition

The product of a non-zero rational number and an irrational number isirrational.

Proof.

Q.E.D.



Finding Mistakes in Proofs

1. a = b Given.2. a2 = ab Multiply both sides by a.3. a2 − b2 = ab− b2 Subtract b2 from both sides.4. (a− b)(a + b) = b(a− b) Factor.5. a + b = b Cancel a− b on both sides.6. 2b = b Substitute a for b since a = b.7. 2 = 1 Divide both sides by b.

Where is the mistake in this proof?



Other Errors in Proofs

Begging the Question

This occurs when the part of the proof is based on the truth of thestatement being proved (we saw this in the previous section).

Circular Reasoning

Occurs when you use a statement to prove itself.


Chapter 1 - Foundations 1.8 - Proof Methods and Strategy

Proof Methods

Proof by Cases

Logically, it has a form similar to:

p ∨ q ∨ rp→ sq → sr → s

∴ sWe can use this method when there are finitely many possibilities andshow that each possible case leads to the desired result.

Exhaustive ProofSimilar to a proof by cases. Generally used when looking at a relativelysmall number of examples will exhaust all possibilities.



Something so important it gets its own frame.

NoteNeither a Proof by Cases nor an Exhaustive Proof will constitute avalid proof unless all cases or possibilities have been examined! “Proofby Life Savers Gummies” (a.k.a., “Proof by doing a couple ofexamples”) is not a valid form of proof!



Example

Theorem (Triangle Inequality)

For any two real numbers x and y, |x + y| ≤ |x|+ |y|.



Existence Proofs

A proof of a proposition of the form ∃xP (x) is called an existenceproof. There are two types of existence proofs:

1. Constructive:

2. Nonconstructive:



Examples

Constructive Example

Show that there is a positive integer that can be written as the sum ofcubes in two different ways.

Nonconstructive Example

Show that there exists two irrational numbers x and y such that xy isrational.



Uniqueness

DefinitionThe Uniqueness Quantifier ∃!xP (x) means

Example

∀y ∈ R (y 6= 0→ ∃!x(xy = 1))



Counterexamples

To show that ∀xP (x) is false it is sufficient to find one value of x forwhich P (x) is false.

Example

Every positive integer is the sum of three squares.



Open Problems

The 3x+ 1 Conjecture

Starting with any positive integer and repeatedly applying thetransformation whereby an even integer gets divided by 2, and an oddinteger gets multiplied by 3 and incremented by 1, we will ultimatelygenerate the integer 1.

Goldbach’s Conjecture

Every positive even integer n ≥ 4 can be written as the sum of twoprime numbers.



Additional Logic Puzzles I

1. Use all of the following premises to reach a conclusion:I The only books in this library, that I do not recommend for

reading, are unhealthy in tone.I The bound books are all well written.I All the romances are healthy in tone.I I do not recommend you to read any of the unbound books.

2. 4Use all of the following premises to reach a conclusion:I All my sons are slim.I No child of mine is healthy who takes no exercise.I All gluttons, who are children of mine, are fat.I No daughter of mine takes any exercise.



Additional Logic Puzzles II

3. Let us assume that there are five houses of different colors next toeach other on the same road. In each house lives a man of adifferent nationality. Every man has his favorite drink, his favoritebrand of cigarettes, and keeps pets of a particular kind.

I The Englishman lives in the red house.I The Swede keeps dogs.I The Dane drinks tea.I The green house is just to the left of the white one.I The owner of the green house drinks coffee.I The Pall Mall smoker keeps birds.I The owner of the yellow house smokes Dunhills.I The man in the center house drinks milk.I The Norwegian lives in the first house.I The Blend smoker has a neighbor who keeps cats.I The man who smokes Blue Masters drinks bier.I The man who keeps horses lives next to the Dunhill smoker.



Additional Logic Puzzles III

I The German smokes Prince.I The Norwegian lives next to the blue house.I The Blend smoker has a neighbor who drinks water.

Who keeps fish as his pet?5



Additional Logic Puzzles IV

4. The Lady or the Tiger.6 A certain king likes to entertainhimself by making his prisoners play a game to decide their fate.The prisoners are presented with two doors. In a room behindeach door is either a lady whom the prisoner may marry, or a tigerwhom may eat the prisoner. A clue is written on each door andthe prisoner decides which door to open based on these clues. Theclues provided to three prisoners brought before the king arebelow. Try to figure out which door each prisoner should open.Prisoner 1 is told that exactly one of the following clues is trueand exactly one is false.

Door 1: There is a lady behind this door and a tiger behindthe other.

Door 2: There is a lady behind one of the doors and a tigerbehind the other.



Additional Logic Puzzles V

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 ladybehind the second door.

Door 2: There is a lady behind this door.

Prisoner 3 receives directions which are a bit tricker since the firsttwo escaped. This prisoner is told that if a lady is behind door 1then the clue on door 1 is true, but if a tiger is behind door 1 thenthe clue on that door is false. Door 2 follows the opposite rule: if alady is behind door 2 the clue on door 2 is false, but if a tiger isbehind 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.

4Puzzles 1 and 2 are attributed to Lewis Carroll5Commonly attributed to Albert Einstein.6Commonly attributed to Raymond Smullyan.


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