Proofs Induction

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Module #1 - Logic

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Basic Proof Methods


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Nature & Importance of Proofs

In mathematics, aproof is an argument that is correct(well-reasoned, logically valid) and

complete(clear, detailed),

so that it rigorously & undeniably establishes thetruth of a mathematical statement.

Why must the argument be correct & complete? Correctnessprevents us from fooling ourselves.

Completenessallows anyone to verify the result. In this course (& throughout mathematics),

a very high standardfor correctness andcompleteness of proofs is demanded!!


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Overview

Methodsof mathematical argument(i.e., proof methods) can be formalized

in terms ofrules of logical inference. Mathematical proofscan themselves be

represented formally as discrete structures.

We will review both correct& fallaciousinference rules, & several proof methods.


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Applications of Proofs

An exercise in clear communication of logicalarguments in any area of study.

The fundamental activity of mathematics is the

discovery and elucidation, through proofs, ofinteresting new theorems.

Theorem-proving has applications in programverification, computer security, automated reasoning

systems, etc. Proving a theorem allows us to rely upon on its

correctness even in the most critical scenarios.


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Proof Terminology

Theorem

A statement that has been proven to be true.

Axioms, postulates, hypotheses, premises

Assumptions (often unproven) defining thestructures about which we are reasoning.

Rules of inference Patterns of logically valid deductions from

hypotheses to conclusions.


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More Proof Terminology

Lemma - relatively uninteresting statements thatare proved on the way (as stepping-stone) toproving an actual theorem of interest

Corollary- A minor theorem proved as an easyconsequence of a major theorem.

Conjecture- A statement whose truth value hasnot been proven. (A conjecture may be widely

believed to be true, regardless.) Theory The set of all theorems that can be

proven from a given set of axioms.


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Graphical Visualization

Various TheoremsThe Axiomsof the Theory

A Particular Theory

A proof


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Inference Rules - General Form

An Inference Ruleis

A pattern establishing that if we know that a

set ofantecedentstatements of certain formsare all true, then we can validly deduce that acertain relatedconsequentstatement is true.

antecedent 1antecedent 2

consequent means therefore


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Inference Rules & Implications

Each valid logical inference rulecorresponds to an implication that is atautology.

antecedent 1 Inference ruleantecedent 2

consequent

Corresponding tautology:((ante. 1) (ante. 2) ) consequent


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Some Inference Rules

p Rule ofAddition pq

pq Rule ofSimplification

p p Rule ofConjunction

q

pq pq Rule ofResolution

pr qr


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Modus Ponens & Tollens

p Rule ofmodus ponenspq (a.k.a. law of detachment)

q q

pq Rule ofmodus tollens

p

the mode ofaffirming

the mode of denying


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Syllogism Inference Rules

pq Rule of hypotheticalqr syllogism

pr p q Rule of disjunctive

p syllogism

q

Aristotle

(ca. 384-322 B.C.)


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Formal Proofs A formal proof of a conclusionC,

given premisesp1, p2,,pnconsists of a sequence ofsteps,each of which

applies some inference ruleto premises or previously-proven statements(antecedents)

to yield a new true statement (theconsequent). A proof demonstrates that

if the premises are true,

then the conclusion is true.


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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.


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Proof Examplecont.

Let us adopt the following abbreviations:

sunny=It is sunny; cold=I t is cold;

swim=We will swim; canoe=We willcanoe; early=We will be home early.

Then, the premises can be written as:It is not sunny and it is cold. (1) sunny cold

We will swim only if it is sunny. (2) swim sunnyIf we do not swim, then we will canoe. (3) swim canoeIf we canoe, then we will be home early. (4) canoe early


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Proof Examplecont.

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.


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Inference Rules for QuantifiedStatements

xP(x)P(o) (substituteanyspecific objecto)

P(g) (for gageneral element of u.d.)xP(x)

xP(x)

P(c) (substitute anewconstantc) P(o) (substitute any extant objecto)

xP(x)


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Examples on Inference Rules for

Quantifiers Ex1:

Show that the statementsEveryone in this computational structures class hastaken a course in computer science andMarla is a student in this class.imply that Marla has taken a course in computer science.

This is universal instantiation Let D(x) denote x is in this computational structures

class, andlet C(x) denote x has taken a course in computerscience.

Then the premises arex (D(x) C(x)) and D(Marla).

The conclusion is C(Marla).


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Examples on Inference Rules for

Quantifiers (cont.) The following steps can be used to establish the

conclusion from the premises.

Step Reason

1. x (D(x) C(x)) Premise

2. D(Marla) C(Marla) universal instantiationfrom (1)

3. D(Marla) Premise

4. C(Marla) Modus ponens from (2) and (3)


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Examples on Inference Rules forQuantifiers (cont.) Ex2:

A student in this class has not read the book andEveryone in this class has passed the first examimply the conclusion

Someone who passed the first exam has not read thebook. This is existential generalization Let C(x) be x is in this class,

B(x) be x has read the book, andP(x) be x passed the first exam.

The premises are:x (C(x) B(x)) andx (C(x) P(x)).

The conclusion is x (P(x) B(x)).


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Examples on Inference Rules forQuantifiers (cont.)

The following steps can be used to establish the conclusionfrom the premises.Step Reason

1. x (C(x) B(x)) Premise2. C(a) B(a) Existential instantiation from (1)3. C(a) Simplification from (2)4. x (C(x) P(x)) Premise5. C(a) P(a) Universal instantiation from (4)

6. P(a) Modus ponens from (3) and (5)7. B(a) Simplification from (2)8. P(a) B(a) Conjunction from (6) and (7)9. x (P(x) B(x)) Existential generalization


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Common Fallacies

A fallacyis an inference rule or other proofmethod that is not logically valid.

A fallacy may yield a false conclusion!

Fallacy ofaffirming the conclusion:

pq is true, andq is true, sopmust be true.(No, becauseFT is true.)

Ex1: Is the following statement valid?

If you do every problem in this book, then you willget an A from this course. Now you got an A, so itmeans you did every problem in this book.

NO

[((pq) q) p] is NOT a tautology!


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Common Fallacies (cont.)

Fallacy ofdenying the hypothesis:

pq is true, andp is false, soqmust be

false. (No, again becauseFT is true.) Ex2: Is the following statement valid?

If you do every problem in this book, then youwill get an A from this course. Since you didnot do all the problems in this book, so youwill not get an A from this course.

NO

[((pq) p) q] is NOT a tautology!


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Circular Reasoning The fallacy of (explicitly or implicitly) assuming

the very statement you are trying to prove in thecourse of its proof. Example:

Prove that ifn2 is even, then n is even.

Attempted proof: Assumen2 is even. Thenn2=2k, for some integer k. Dividing both sides by ngivesn= (2k)/n= 2(k/n). So there is an integerj

(namely k/n) such thatn=2j. Thereforen is even. Circular reasoning is used in this proof. Where?

Begs the question: How doyou show thatj=k/n=n/2 is an integer,

without firstassuming thatnis even?


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Proof Methods for Implications

For proving implicationspq, we have: Directproof: Assumep is true, and proveq.

Indirectproof: Assumeq, and provep.


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Direct ProofExample

Definitions:An integer n is calledodd iff n=2k+1for someinteger k;

n iseven iff n=2k for some k. Axiom: Every integer is either odd or even. Theorem: (For all numbersn)

Ifn is an odd integer, thenn2 is an odd integer.

Proof: Ifn is odd, thenn= 2k+1for some integer k.Thus, n2 = (2k+1)2 = 4k2 + 4k+ 1 = 2(2k2 + 2k) + 1.Thereforen2 is of the form2j + 1(withj the integer 2k2 + 2k),thusn2 is odd.


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Indirect ProofExample Theorem: (For all integersn)

If3n+2is odd, thenn is odd.

Proof:

Suppose that the conclusion is false, i.e., thatn is even.Thenn=2k for some integer k.Then3n+2 = 3(2k)+2 = 6k+2 = 2(3k+1).Thus3n+2is even, because it equals2j for integerj =3k+1. So3n+2is 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.


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Proof byContradiction

A method for provingp.

Assumep, andprove both qandq,for some propositionq. (Can be anything!)

Thusp (q q)

(q q) is a trivial contradiction, equal toF

ThuspF,which is true only ifp=F

Thusp is true.


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Proof by Contradiction Example

Theorem: 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.Thusj2 is even, soj is even.But then i andj have a common divisor,namely 2,

so we have a contradiction.

2


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Proving Existentials

A proof of a statement of the formxP(x)is called anexistence proof.

If the proof demonstrates how to actuallyfind or construct a specific elementa,such thatP(a) is true,

then it is aconstructiveproof. Otherwise, it isnonconstructive.


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Constructive Existence Proof

Theorem:There exists a positive integer nthat is the sum

of two perfect cubes in two different ways: equal toj3 +k3 and l3 +m3, wherej, k, l, mare

positive integers, and {j,k} {l,m}

Proof: Considern= 1729,j = 9, k= 10, l = 1, m= 12.Now just check that the equalities hold.


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Nonconstructive Existence Proof Show that there exist irrational numbersxandy

such thatxyis rational.

Proof: We know that is irrational (accept it).

If is rational, choose x= andy= .

Then, xy is (rational).

If is irrational, choose x= andy= .Then, xy is 2 (rational).

We showed that in both cases the result is rational,

without knowing which of the pairs of x and y works..

2

2

2

2 2

2

2 2

2

2

2

2


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Review: Proof Methods Direct, indirectproofs of statements of the

formpq. Proof bycontradictionof any statements.

Constructiveandnonconstructiveexistence

proofs.

Module#1 Logic


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Module #1 - Logic

34

Induction


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Topics

Mathematical induction

Well ordered property

Second principle of induction


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Mathematical Induction

A powerful, rigorous technique for provingthat a predicateP(n) is true for every

natural number n, no matter how large. Essentially a domino effect principle.

Based on a predicate-logic inference rule:

P(0)n0(P(n)P(n+1))n0P(n)


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Validity of Induction

Proof that k0P(k) is a valid consequent:

Given anyk0, n0(P(n)P(n+1)) (antecedent 2)trivially impliesn0 (n


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The Well-Ordering Property

The validity of the inductive inference rule can also beproved using thewell-ordering property, which says:

Every non-empty set of non-negative integers has aminimum (smallest) element.

SN : mS : nS : mn

Assume that there exists at least one positive integer

n: {n|P(n)}. This non-empty set has a minimumelementm. Som-1 is not in this set and thusP(m-1) istrue, but thenP(m-1)P((m-1)+1) contradicted withP(m).

Hence, P(n) must be true non-negative integers


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Outline of an Inductive Proof

Want to provenP(n)

Base case (or basis step): Prove P(0).

Inductive step: Proven P(n)P(n+1). E.g. use a direct proof:

LetnN, assumeP(n). (inductive hypothesis)

Under this assumption, proveP(n+1). Inductive inference rule then givesn

P(n).


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Generalizing Induction

Can also be used to provencP(n) for agiven constantcZ, where maybec0.

In this circumstance, the base case is to proveP(c) rather thanP(0), and the inductive step

is to provenc (P(n)P(n+1)).


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Second Principle of Induction

Characterized by another inference rule:P(0)

n0: (0knP(k)) P(n+1)n0: P(n)

Difference with 1st principle is that the

inductive step uses the fact thatP(k) is truefor all smallerk


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Induction Example (1st princ.)

Prove that the sum of the firstnodd positiveintegers isn2. That is, prove:Proof by induction.

Base case: Letn=1. The sum of the first 1 oddpositive integer is 1 which equals 12.

Inductive step: Proven1: P(n)P(n+1).

Letn1, assumeP(n), and proveP(n+1).

2

1

)12(:1 ninn

i

= =

2

2

1

1

1

)1(

12

)1)1(2()12()12(

+=

++=

++

=

=

+

=

n

nn

nii

n

i

n

i


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Another Induction Example

Prove that n>0, n


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Example of Second Principle

Show that everyn>1 can be written as a productp1p2ps of some series ofsprime numbers. LetP(n)=nhas that property

Base case: n=2, lets=1, p1=2.

Inductive step: Letn2. Assume2kn: P(k).Considern+1. If prime, lets=1, p1=n+1.

Elsen+1=ab, where 2anand 2bn.Sincea=p1p2pt andb=q1q2qu, thenn+1=p1p2ptq1q2qu, a product ofs=t+uprimes.


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Another 2nd Principle Example

Prove that every amount of postage of 12cents or more can be formed using just 4-

cent and 5-cent stamps. Base case: 12=3(4), 13=2(4)+1(5),

14=1(4)+2(5), 15=3(5), so12n15,P(n).

Inductive step: Letn15, assume12knP(k). Note 12n3n, soP(n3), so add a4-cent stamp to get postage for n+1.

Proofs Induction

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