Chapter 1: The Foundations: Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers

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Chapter 1: The Foundations: Logic and Proofs

1.1 Propositional Logic1.2 Propositional Equivalences1.3 Predicates and Quantifiers1.4 Nested Quantifiers1.5 Rules of Inference1.6 Introduction to Proofs1.7 Proof Methods and Strategy

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1.1: Propositional Logic

Propositions: A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.

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Example 1: All the following declarative sentences are propositions: 1. Washington D.C., is the capital of

the USA.2. Toronto is the capital of Canada3. 1+1=2.4. 2+2=3.

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Example 2: Consider the following sentences. Are they propositions? 1. What time is it?2. Read this carefully.3. x+1=2.4. x+y=z

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• We use letters to denote propositional variables (or statement variables).

• T: the value of a proposition is true.• F: the value of a proposition is false.• The area of logic that deals with

propositions is called the propositional calculus or propositional logic.

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Let p and q are propositions:

Definition 1: Negation (Not)• Symbol: ¬ • Statement: “it is not the case that p”.

• Example:P: I am going to town

¬P:It is not the case that I am going to town;I am not going to town;I ain’t goin’.

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Definition 2: Conjunction (And)• Symbol:• The conjunction pq is true when both p and q are

true and is false otherwise.

• Example:

P - ‘I am going to town’Q - ‘It is going to rain’PQ: ‘I am going to town and it is going to rain.’

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Definition 3: Disjunction (Or)• Symbol:• The disjunction pq is false when both p and q are

false and is true otherwise.

• Example:

P - ‘I am going to town’Q - ‘It is going to rain’P Q: ‘I am going to town or it is going to rain.’

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Definition 4: Exclusive OR• Symbol:• The exclusive or of p and q, denote pq, is true

when exactly one of p and q is true and is false otherwise.

• Example:

P - ‘I am going to town’Q - ‘It is going to rain’P Q: ‘Either I am going to town or it is going to rain.’

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Definition 5: Implication • If…. Then….• Symbol:• The conditional statement pq is false when p is true

and q is false, and true• P is called the hypothesis and q is called the

conclusion.• Example:

P - ‘I am going to town’Q - ‘It is going to rain’P Q: ‘If I am going to town then it is going to rain.’

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Equivalent Forms

If P, then QP implies Q

If P, QP only if Q

P is a sufficient condition for QQ if P

Q whenever PQ is a necessary condition for P

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Note: The implication is false only when P is true and Q is false!• ‘If the moon is made of green cheese then I have

more money than Bill Gates’ (?)• ‘If the moon is made of green cheese then I’m on

welfare’ (?)• ‘If 1+1=3 then your grandma wears combat boots’ (?) • ‘If I’m wealthy then the moon is not made of green cheese.’ (?)• ‘If I’m not wealthy then the moon is not made of

green cheese.’ (?)

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More terminology

•QP is the CONVERSE of P Q•¬ Q ¬ P is the CONTRAPOSITIVE of P Q•¬ P ¬ Q is the inverse of P Q•Example:Find the converse of the following statement:R: ‘Raining tomorrow is a sufficient condition for my not going to town.’

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ProcedureStep 1: Assign propositional variables to component propositions P: It will rain tomorrow Q: I will not go to townStep 2: Symbolize the assertion R: P QStep 3: Symbolize the converse Q PStep 4: Convert the symbols back into words ‘If I don’t go to town then it will rain tomorrow’

• Homework: Find inverse and contrapositive of statements above.

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Definition 6: Biconditional• ‘if and only if’, ‘iff’ • Symbol:• The biconditional statement pq is true when p and q

have the same truth value, and is false otherwise. • Biconditional statements are also called bi-implications.• Example:

P - ‘I am going to town’Q - ‘It is going to rain’P Q: ‘I am going to town if and only if it is going to rain.’

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Translating English•Breaking assertions into component propositions - look for the logical operators!•Example:‘If I go to Harry’s or go to the country I will not goshopping.’P: I go to Harry’sQ: I go to the countryR: I will go shoppingIf......P......or.....Q.....then....not.....R(P Q) ¬ R

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Constructing a truth table1. one column for each propositional variable2. one for the compound proposition3. count in binary4. n propositional variables = 2n rows

• Construct the truth table for (P ¬ Q) (PQ)• HW: Construct the truth table for (P Q) ¬ R

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• What is the real meaning of ¬ PQ ? a) (¬ P) Q b) ¬ (PQ)• What is the real meaning of PQR ? a) (PQ)R b) P(QR) • What is the real meaning of P QR ? a) (P Q)R b) P (QR)

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• Example 20Find the bitwise OR, bitwise AND, and

bitwise XOR of the bit strings 01 1011 0110 and 11 0001 1101.

Logic and Bit Operations

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• Example 18: There are two kind of inhabitants, knights,

who always tell the truth, and their opposites, knaves, who always lie. You encounter two people A and B. What are A and B if A says “B is a knight” and B says “The two of us are opposite type”?

Logic Puzzles

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• Proposition• Negation• Conjection• Disjunction• Exclusive OR• Implication

Terms• Inverse• Converse• Contrapositive



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1.2: Propositional EquivalencesDefinition:Tautology: A compound proposition that

is always true.Contradiction: A compound proposition

that is always false.Contingency: A compound proposition

that is neither a tautology nor a contradiction.

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Logical Equivalences• Compound propositions that have the

same truth values in all possible cases are called logically equivalent.

• Definition:The compound propositions p and q are

called logically equivalent if pq is a tautology. Denote pq.

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Logical Equivalences• One way to determine whether

two compound propositions are equivalent is to use a truth table.

• Symbol: PQ

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Logical Equivalences• Prove the De Morgan’s Laws.

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• HW: Prove the other one (De Morgan’s Laws).

Logical Equivalences

•Example: Show that pq and ¬pq are logically equivalent. •HW: example 4 of page 23

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Example 5: Use De Morgan’s laws to express the negations of “Miguel has a cellphone and he has a laptop computer”.

Example 5: Use De Morgan’s laws to express the negations of “Heather will go to the concert or Steve will go to the concert”.

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• Example 6: Show that ¬(pq) and p ¬q are logically equivalent.

• Example 7: Show that ¬(p(¬p q)) and ¬p ¬q are logically equivalent by developing a series of logical equivalences.

• Example 8: Show that (p q) ( pq) is a tautology.

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• Tautology• Contradiction• Contingency• Logical Equivalence• De Morgan’s Laws• Commutative Law• Associative Law• Distributive Law

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Terms



Predicate: A generalization of propositions ; A propositions which contain variables

• Predicates become propositions once every variable is bound- by– assigning it a value from the Universe of

Discourse U or–quantifying it

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Predicates

Examples:Let U = Z, the integers = {. . . -2, -1, 0 , 1, 2, . . .}– P(x): x > 0 is the predicate. It has no truth value until

the variable x is bound.• Examples of propositions where x is assigned a

value:(a) P(-3) (?, true or false); (b) P(0)(?); (c) (c) P(3)(?).

• The collection of integers for which P(x) is true are the positive integers.

• P(y) ν ¬ P(0) is not a proposition. The variable y has not been bound. However, P(3) ν ¬ P(0) is a proposition which is true. P. 1

Predicates

Example: Let R be the three-variable predicate R(x, y, z): x + y = z

• Find the truth value of R(2, -1, 5), R(3, 4, 7), R(x, 3, z)

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Predicates

Quantifiers: Universal

• P(x) is true for every x in the universe of discourse.

• Notation: universal quantifier ∀xP(x)• ‘For all x, P(x)’, ‘For every x, P(x)’• The variable x is bound by the universal

quantifier producing a proposition.• An element for which P(x) is false is

called a counterexample of ∀xP(x).• Example: U={1,2,3}

∀ xP(x) P(1) Λ P(2) Λ P(3)P. 1

Quantifiers: Universal

• Example 8: Let P(x) be the statement “x+1>x.” What is the

truth value of the quantification xP(x) where ∀the domain consists of all real number.

• HW: P36, example 13

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Quantifiers: Existential

• P(x) is true for some x in the universe of discourse.

• Notation: existential quantifier ∃xP(x)– ‘There is an x such that P(x),’ – ‘For some x, P(x)’, – ‘For at least one x, P(x)’, – ‘I can find an x such that P(x).’

• Example: U={1,2,3}–∃xP(x) P(1) ν P(2) ν P(3)

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Quantifiers: Existential

• Example 14: Let P(x) denote the statement “x>3.” What is the

truth value of the quantification xP(x), ∃where the domain consists of all real numbers.

• HW: Page 37, Example 16.

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Quantifiers

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Quantifiers: Unique Existential

• P(x) is true for one and only one x in the universe of discourse.

• Notation: unique existential !∃ xP(x)– ‘There is a unique x such that P(x),’– ‘There is one and only one x such that P(x),’– ‘One can find only one x such that P(x).’

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Quantifiers

• Example: U={1,2,3}• Truth Table:

P(1) P(2) P(3) !∃ xP(x) 0 0 0 00 0 1 10 1 0 10 1 1 01 0 0 11 0 1 01 1 0 01 1 1 0

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Quantifiers

• Note: A predicate is not a proposition until all variables have been bound either by quantification or assignment of a value!

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Precedence of Quantifiers

The quantifiers and have higher precedence than all logical operators from propositional calculus.

Example: xP(x)Q(x) means(a) ( xP(x))Q(x) (b) x(P(x)Q(x))

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Binding Variables

• Example:x(x+y=1)Bound variable:Free variable:

• Example:x(P(x)Q(x)) xR(x)The scope of :The scope of :The same meaning of x(P(x)Q(x)) yR(y)

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Logical Equivalences Involving Quantifiers

Example 19: Show that x(P(x)Q(x)) and xP(x) xQ(x) are

logically equivalent.

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Logical Equivalences Involving Quantifiers

• Statement involving predicates and quantifiers are logically equivalent if and only if they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for the variables in these propositional functions.

• Symbol: S T

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Equivalences Involving the Negation Operator

• Equivalences involving the negation operator¬∀xP(x) ∃x ¬ P(x)¬∃xP(x) ∀ x ¬ P(x)

• Distributing a negation operator across a quantifier changes a universal to an existential and vice versa.

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Equivalences Involving the Negation Operator

• Example 20:What are the negations of the statements: “There is an honest politician”

• Example 20:What are the negations of the statements: “All Americans eat cheeseburgers”.

HW: Example 21 in P41.

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Translating from English into Logical Expressions (single quantifier)

• Example 23:Express the statement “Every student in this class has

studied calculus” using predicates and quantifiers.

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Using Quantifiers in System Specifications

• Example 26:Consider these statements. “All lions are fierce.”“Some lions do not drink coffee.”“Some fierce creatures do not drink coffee.”Let P(x): x is a lion. Q(x): x is fierce. R(x): x drinks coffee.Assuming that the domain consists of all creatures,

express the statements in the argument using quantifiers and P(x), Q(x), and R(x).

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Logic Programming

• Prolog(Programming in Logic, developed in the 1970s.

• Working in AI.• Prolog programs including a set of declarations

consisting of two types of statements:– Facts : define predicates by specifying the elements that

satisfy these predicates.– Rules: define new predicates using those already defined

by facts.

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Logic Programming• Example 28:Facts:instructor(chan, math273)instructor(patel, ee222)instructor(grossman, cs301)enrolled(kevin, math273)enrolled(juana, ee222)enrolled(juana, cs301)enrolled(kiko, math273)enrolled(kiko, cs301)

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Rules:teacher(P,S) :-

instructor(P,C), enrolled(S,C)

Queries:?enrolled(kevin, math273)?enrolled(X,math273)?teacher(X,juana)

Uppercase letters are variables. The “” represents by “,” and the “” represents by “;” in Prolog.

Quantifiers

• Multiple Quantifiers: read left to right . . .• Example: Let U = R, the real numbers, P(x,y): xy= 0

∀x∀yP(x, y)∀x∃yP(x, y)∃x∀yP(x, y)∃x∃yP(x, y)

The only one that is false is the first one.• Suppose P(x,y) is the predicate x/y=1? Assume U=R-

{0}.

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Quantifiers

• Example: Let U = {1,2,3}. Find an expression equivalent to ∀x∃yP(x, y) where the variables are bound by substitution instead:

• Expand from inside out or outside in.• Outside in:

∃yP(1, y) Λ ∃ yP(2, y) Λ ∃ yP(3, y)[P(1,1) ν P(1,2) ν P(1,3)] Λ[P(2,1) ν P(2,2) ν P(2,3)] Λ[P(3,1) ν P(3,2) ν P(3,3)]

• HW: Inside Out: P. 1

Quantifiers

• De Morgan’s Laws for Quantifiers

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Quantifiers: Converting from English

• Examples:F(x): x is a fleegleS(x): x is a snurdT(x): x is a thingamabob

U={fleegles, snurds, thingamabobs}• Note: the equivalent form using the existential

quantifier is also given

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• Everything is a fleegle∀xF( x) ¬∃x¬F(x)

• Nothing is a snurd.∀x¬S(x) ¬∃xS( x)

• All fleegles are snurds.∀x[F(x) → S(x)]

Û ∀ x[¬ F(x) ν S(x)] ∀ x ¬[F(x) Λ ¬ S(x)] ¬∃x[F(x) Λ ¬ S( x)]

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• Some fleegles are thingamabobs.∃x[F(x) Λ T(x)]

¬ ∀ x[¬ F(x) ν ¬ T(x)]• No snurd is a thingamabob.

∀ x[S(x) → ¬ T(x)] ¬ ∃ x[S(x ) Λ T(x)]

• If any fleegle is a snurd then it's also a thingamabob ∀ x[(F(x) Λ S(x)) → T(x)]

¬ ∃ x[F(x) Λ S(x) Λ ¬ T( x)]

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Quantifiers:Dangerous situations

• Commutativity of quantifiers∀x∀yP(x, y) ∀y∀xP(x, y)? YES!∀x∃yP(x, y) ∃y∀xP(x, y)? NO!

DIFFERENT MEANING!• Example: P(x,y): x+y=0, U: Integers∀x∃yP(x, y) is Ture∃y∀xP(x, y) is False

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Quantifiers:Dangerous situations

• Distributivity of quantifiers over operators∀x[P(x)ΛQ(x)] ∀xP( x)Λ∀xQ(x)? YES!

∀x[P(x)→Q(x)] [∀xP(x)→∀xQ(x)]? NO!• Let P(x) sometimes true, sometimes false, and

Q(x) is always false, then ∀x[P(x)→Q(x)] is False

[∀xP(x)→∀xQ(x)] is True

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Terms

• Proposition• Predicate• Universal Quantifier• Existential Quantifier• Unique Existential Quantifier• De Morgan’s Laws for Quantifiers• Binding Variables• Logic Programming

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Introduction: Nested Quantifiers

• Nested quantifiers: Two quantifiers are nested if one is within the scope of the other.

• Example: xy(x+y=0)

Introduction: Nested Quantifiers• Example:Domain: real number.Addition inverse: xy(x+y=0)Commutative law for addition: xy(x+y=y+x)Associative law for addition: xyz (x+(y+z)=(x+y)

+z)

The Order of Quantifiers• Example:Let Q(x,y) denote “x+y=0.” What are the truth

values of the quantifications yxQ(x,y) and xyQ(x,y), where the domain for all variables consists of all real numbers?

Translating Mathematical Statements into Statements Involving Nested Quantifiers• Example 7:Translate the statement “Every real umber except

zero has a multiplicative inverse.” (A multiplicative inverse of a real number x is a real number y such that xy=1)

• HW: Example 8, p54

Translating from Nested Quantifiers into English

• Example 9:Translate the statement x(C(x)y(C(y)F(x,y)))

into English, where C(x) is “x has a computer,” F(x,y) is “x and y are friends,” and the domain for both x and y consists of all students in your school.

• HW: Example 10, p55.

Negating Nested Quantifiers• Example 14:Express the statement xy(xy=1) negation of the

statement so that no negation precedes a quantifier.

• HW: Example 15, p57



• Definition:An argument in propositional logic is a sequence of

propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true.

Rules of Inference

• Definition: A theorem is a valid logical assertion which can be proved using– other theorems– axioms (statements which are given to be true) and– rules of inference (logical rules which allow the deduction

of conclusions from premises).

• A lemma (not a “lemon”) is a 'pre-theorem' or a result which is needed to prove a theorem.

• A corollary is a 'post-theorem' or a result which follows directly from a theorem.

Rules of Inference

• Many of the tautologies in Chapter 1 are rules of inference. They have the form

H1 Λ H2 Λ..... Λ Hn →C• Where Hi are called the hypotheses and C is the conclusion.• As a rule of inference they take the symbolic form:

H1

H2

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.Hn

Cwhere means 'therefore' or 'it follows that.'

Rules of Inference

• Examples:The tautology P Λ(P → Q) → Q becomesP

P → Q Q

• This means that whenever P is true and P → Q is true we can conclude logically that Q is true.

• This rule of inference is the most famous and has the name– modus ponens or– the law of detachment.

Rules of Inference

Rules of Inference

Rules of Inference for Quantifiers

• Note:– In Universal Generalization, x must be arbitrary.– In Universal Instantiation, c need not be arbitrary but often is assumed to be.– In Existential Instantiation, c must be an element of the universe which makes P(x)

true.

• Example: Every man has two legs. John Smith is a man. Therefore, John Smith has two legs.

• Define the predicates:M(x): x is a man

L(x): x has two legsJ: John Smith, a member of the universe

Rules of Inference

• Example: • The argument becomes

1.x[M(x) → L(x)]2.M( J )\ L( J)

• The proof is1. x[M(x) → L(x)] Hypothesis 12.M( J ) → L(J ) step 1 and UI3.M( J ) Hypothesis 24.L( J) steps 2 and 3 and modus ponensQ. E. D. Note: Using the rules of inference requires lots of practice.

Rules of Inference

• Example 12: Show that the premises “Everyone in this

discrete mathematics class has taken a course in computer science” and “Marla is a student in this class” imply the conclusion “Marla has taken a course in computer science.”

• HW: Example 13, p71.

Rule of Inference for Quantified Statement

• Fallacies are incorrect inferences.• Some common fallacies:–The Fallacy of Affirming the

Consequent–The Fallacy of Denying the Antecedent

(or the hypothesis)–Begging the question or circular

reasoning

Fallacies

• Example:If the butler did it he has blood on his hands.The butler had blood on his hands.Therefore, the butler did it.

• This argument has the formP → Q

QPor

[(P → Q)ΛQ] → P• which is not a tautology and therefore not a rule of

inference!

Fallacies:The Fallacy of Affirming the Consequent

• Example:if the butler is nervous, he did it.The butler is really mellow.Therefore, the butler didn't do it.

• This argument has the formP → Q

¬ P ¬ Q

or[(P → Q)Λ ¬ P] → ¬ Q

• which is also not a tautology and hence not a rule of inference.

Fallacies:The Fallacy of Denying the Antecedent (or the hypothesis)

Fallacies: Begging the question or circular reasoning

• This occurs when we use the truth of statement being proved (or something equivalent) in the proof itself.

• Example: Conjecture: if x2 is even then x is even.Proof: If x2 is even then x2 = 2k for some k. Then

x = 2l for some l. Hence, x must be even.

• Example: P (x): x注射疫苗 Q(x): x死亡某報刊登：『某人於下午過世，上午有注射疫苗』，請問其論點及目的為何？

Example:上題中加入一個描述：”所有注射疫苗的人都會死亡”。某人不幸過世，請問他是否有打疫苗？

Example:要如何確認 (加入 rules)某人是因注射疫苗過世。

HW: Example 10, p69

Fallacies

• Argument• Premises• Conslusion• Valid

Terms

• Theorem• Axioms• Lemma• Corollary

• Rule of inference• Modus ponens• Modus tollensva• fallacies



• To prove an argument is valid or the conclusion follows logically from the hypotheses:– Assume the hypotheses are true– Use the rules of inference and logical

equivalences to determine that the conclusion is true.

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Formal ProofsFormal Proofs

Example:• Consider the following logical argument:

If horses fly or cows eat artichokes, then the mosquito is the national bird. If the mosquito is the national bird then peanut butter takes good on hot dogs. But peanut butter tastes terrible on hot dogs. Therefore, cows don't eat artichokes.

1. Assign propositional variables to the component propositions in the argument:

F Horses flyA Cows eat artichokesM The mosquito is the national birdP Peanut butter tastes good on hot dogs

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2. Represent the formal argument using the variables

1.(F ν A) → M2.M →P3. ¬ P ¬ A

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3. Use the hypotheses 1., 2., and 3. and the above rules of inference and any logical equivalences to construct the proof.

Assertion Reasons1.(F ν A) → M Hypothesis 1.2.M → P Hypothesis 2.3.(F ν A) → P` steps 1 and 2 and hypothetical

syll.4. ¬ P Hypothesis 3.5. ¬(F ν A) steps 3 and 4 and modus tollens6. ¬F Λ¬A step 5 and DeMorgan7. ¬A Λ¬F step 6 and commutativity of

'and'8. ¬A step 7 and simplification

Q. E. D.P. 1


• We wish to establish the truth of the 'theorem‘

P→Q.• P may be a conjunction of other

hypotheses.• P → Q is a conjecture until a proof is

produced.

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Methods of Proof

• Trivial Proof• Vacuous Proof• Direct Proof• Indirect Proof• Proof by Contradiction• Proof by Cases

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Methods of Proof

• Trivial proof – If we know Q is true then P → Q is true.

• Example:– If it's raining today then the void set is a subset

of every set.– The assertion is trivially true independent of

the truth of P.

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Methods of Proof: Trivial Proof

• Vacuous proof– If we know one of the hypotheses in P is false then P →

Q is vacuously true.

• Example:– If I am both rich and poor then hurricane Fran was a

mild breeze.

• This is of the form(P Λ ¬ P) → Q

– and the hypotheses form a contradiction.– Hence Q follows from the hypotheses vacuously.

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Methods of Proof: Vacuous Proof

• Direct proof– assumes the hypotheses are true– uses the rules of inference, axioms and any

logical equivalences to establish the truth of the conclusion.

• Example:– Theorem: If 6x + 9y = 101, then x or y is not an integer.– Proof: Assume 6x + 9y = 101 is true.

Then from the rules of algebra 3(2x + 3y) = 101.But 101/3 is not an integer so it must be the case that one of 2x or 3y is not an integer (maybe both).Therefore, one of x or y must not be an integer.Q.E.D.

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Methods of Proof: Direct Proof

• Indirect proof– A direct proof of the contrapositive:– assumes the conclusion of P → Q is false (¬ Q is true)– uses the rules of inference, axioms and any logical

equivalences to establish the premise P is false.• Note, in order to show that a conjunction of

hypotheses is false is suffices to show just one of the hypotheses is false.

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Methods of Proof: Indirect Proof

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Methods of Proof: Indirect Proof

• Example:– A perfect number is one which is the sum of all

its divisors except itself. For example, 6 is perfect since 1 + 2 + 3 = 6. So is 28.

– Theorem: A perfect number is not a prime.– Proof: (Indirect). We assume the number p is a

prime and show it is not perfect.But the only divisors of a prime are 1 and itself.Hence the sum of the divisors less than p is 1 which is not equal to p.Hence p cannot be perfect.Q. E. D.

• Proof by contradiction– assumes the conclusion Q is false– derives a contradiction..

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Methods of Proof:Proof by contradiction

• Example:– Theorem: There is no largest prime number.

(Note that there are no formal hypotheses here.)We assume the conclusion 'there is no largest prime

number' is false. There is a largest prime number. Call it p. Hence, the set of all primes lie between 1 and p.

Form the product of these primes:r = 2•3•5•7•11•....•p.But r + 1 is a prime larger than p. (Why?).This contradicts the assumption that there is a largest

prime. Q.E.D.

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• The formal structure of the above proof is as follows:– Let P be the assertion that there is no largest prime.– Let Q be the assertion that p is the largest prime.– Assume ¬ P is true.– Then (for some p) Q is true so ¬ P→Q is true.– We then construct a prime greater than p so Q → ¬ Q.– Applying hypothetical syllogism we get ¬ P → ¬ Q.– From two applications of modus ponens we conclude

that Q is true and ¬ Q is true so by conjunction ¬ QΛQ or a contradiction is true.

– Hence the assumption must be false and the theorem is true.

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1. Break the premise of P→Q into an equivalent disjunction of the form

P1 ν P2 ν... ν Pn .2. Then use the tautology [(P1 → Q) Λ (P2 → Q) Λ... Λ(Pn → Q)]↔[(P1 ν P2 ν... ν Pn )

→ Q] Each of the implications Pi → Q is a case.• You must

a) Convince the reader that the cases are inclusive, i.e., they exhaust all possibilities

b) Establish all implications

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Methods of Proof: Proof by Cases

• Example:– Let be the operation 'max' on the set of

integers:

if a b then a b = max {a, b} = a = b a.– Theorem: The operation is associative.– For all a, b, c

(a b) c = a (b c).

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• Example:Proof:– Let a, b, c be arbitrary integers.– Then one of the following 6 cases must hold (are

exhaustive):1. a b c 2. a c b3. b a c 4. b c a5. c a b 6. c b a

– Case 1: a b = a, a c = a, and b c = b.– Hence (a b) c = a = a (b c).– Therefore the equality holds for the first case.– The proofs of the remaining cases are similar (and are

left for the student).Q. E. D.

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• Example 15:What is wrong with the “proof” that 1=2?“Proof:” We use these steps where a and b are two

equal positive integers.1. a=b (Given)2. a2=ab3. a2-b2=ab-b2

4. (a-b)(a+b)=b(a-b)5. a+b=b6. 2b=b7. 2=1

• HW: Example 15, p83 P. 1

Mistakes in Proofs

• Conjunction• Disjunction• Conjecture

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Terms



• We wish to establish the truth ofxP( x).

• Constructive existence proof:– Establish P(c) is true for some c in the universe.– Then xP( x) is true by Existential Generalization (EG).

• Example:Theorem: There exists an integer solution to the

equation x2+y2=z2.Proof: Choose x = 3, y = 4, z = 5.

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Existence Proofs

• Example:–Theorem: There exists a bijection

from A= [0,1] to B= [0, 2].–Proof:–we could have chosen g(x) = x/2 and

obtained a bijection directly.

• HW: Prove that g(x) above is a bijection function.

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Existence Proofs

• Nonconstructive existence proof.– Assume no c exists which makes P(c) true and derive a

contradiction.• Example:

– Theorem: There exists an irrational number.– Proof:Assume there doesn’t exist an irrational number.Then all numbers must be rational.Then the set of all numbers must be countable.Then the real numbers in the interval [0, 1] is a countable set.But we have already known this set is not countable.Hence, we have a contradiction (The set [0,1] is countable and not countable).Therefore, there must exist an irrational number.

Q. E. D.– Note: we have not produced such a number!

HW: Prove that [0,1] is not countable.

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Nonconstructive Existence Proof

• Recall that x ¬ P(x)↔¬ xP(x ).• To establish that ¬xP(x ) is true (or xP(x) is

false) construct a c such that ¬ P(c) is true or P(c) is false.

• In this case c is called a counterexample to the assertion xP(x)

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Disproof by Counterexample

• Example:Prove or disprove that “every positive integer

is the sum of the squares of two integers.Example: 5=12+22; 34=32+52.Counterexample:

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Disproof by Counterexample

• Nonexistence Proofs– We wish to establish the truth of ¬ xP( x)

(which is equivalent to x ¬ P(x)).– Use a proof by contradiction by assuming there

is a c which makes P(c) true.

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Nonexistence Proofs

• We wish to establish the truth of xP(x) • We assume that x is an arbitrary member of

the universe and show P(x) must be true. Using UG it follows that xP(x) .

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Universally Quantified Assertions

• Example:– Theorem: For the universe of integers, x is even iff

x2 is even.Proof: The quantified assertion is x[x is even ↔ x2

is even]We assume x is arbitrary.Recall that P ↔ Q is equivalent to (P→Q) Λ (Q → P).

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• Example:– Case 1 . We show if x is even then x2 is even using a

direct proofIf x is even then x = 2k for some integer k. Hence, x2 = 4k2 which is even since it is an integer

which is divisible by 2.This completes the proof of case 1.

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• Example:– Case 2 . We show that if x2 is even then x must be even .We use an indirect proof:Assume x is not even and show x2 is not even.If x is not even then it must be odd.So, x = 2k + 1 for some k.Then x2 = (2k+1) 2 = 2(2k 2+2k)+1 which is odd and hence not

even.This completes the proof of the second case.Therefore we have shown x is even iff x2 is even.Since x was arbitrary, the result follows by UG.

Q.E.D.

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• Dear students: Learning how to construct proofs is probably one of the most difficult things you will face in life. Few of us are gifted enough to do it with ease. One only learns how to do it by practicing .

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Rosen to his student

Documents

Chapter 1: The Foundations: Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers