lecture 07© vadim bulitko : cmput 272, winter 2004, uofa1 cmput 272 formal systems logic in cs...

Lecture 07 © Vadim Bulitko : CMPUT 272, Winter 2004, UofA 1

CMPUT 272CMPUT 272Formal Systems Logic in Formal Systems Logic in

CSCS

CMPUT 272CMPUT 272Formal Systems Logic in Formal Systems Logic in

CSCS

Vadim BulitkoUniversity of Alberta

http://www.cs.ualberta.ca/~bulitko/W04


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NumbersNumbersNumbersNumbers

We will work with numbers for the next little whileR is the set of all real numbers

0, 1, -5.27, Pi, …

Q is the set of rational numbers0, 1, -5.27, …

Z is the set of integer numbers0, 1, -1, 2, …

N is the set of natural numbers0, 1, 2, …


Odd / EvenOdd / EvenOdd / EvenOdd / Even

Define predicate even(n):n is an even integer iffExists an integer k such that n=2knZ [even(n) kZ [n=2k]]

Define predicate odd(n):n is an even integer iffExists an integer k such that n=2k+1nZ [odd(n) kZ [n=2k+1]]


A theoremA theoremA theoremA theorem

“Every integer is even or odd. But not both”

nZ [(odd(n) v even(n)) & ~(odd(n) & even(n))]

Need some premises:Integer numbers are closed under +, *Trichotomy law: any number >0, =0, <0Well ordering principle: Every non-empty set of naturals contains its smallest element

Proof – recall how we introduced integers


Lemma 1Lemma 1Lemma 1Lemma 1

Let’s prove an auxiliary statement first:

For any r,xR if 0<r,x<1 then rx<rr,xR [0<r,x<1 rx<r]

Proof:x<1r>0 rx<r


Lemma 2Lemma 2Lemma 2Lemma 2

Let’s prove another auxiliary statement:There are no integers between 0 and 1~ ( nN [0<n<1] )

Proof:Suppose notThen nN [0<n<1]Consider set S={n, n2, n3, …}Every sS is integer and non-negative (i.e., natural)Therefore min(S)=sm S

Consider sm*n then: sm*nS and sm*n<sm (by Lemma 1)Thus, smmin(S) Contradiction


Theorem (part 1)Theorem (part 1)Theorem (part 1)Theorem (part 1)

“Every integer is even or odd”nZ [ odd(n) v even(n) ]

Proof:Suppose not

Then nZ [~(odd(n) v even(n))]

Then let’s find the minimum m=2k such that n<mClearly, 2k-2<n<2kSubtract 2k from both sides: 0<n-2k+2<2Then n-2k+2 is : between 0,1 or between 1,2 or 1


Theorem (part 2)Theorem (part 2)Theorem (part 2)Theorem (part 2)

“No integer is even and odd”~ nZ [ odd(n) & even(n) ]

Proof:Suppose not

Then nZ [odd(n) & even(n)]

By the definitions: n=2k and n=2m+1Thus, 2k=2m+1Subtract 2m from both sides: 2(k-m)=1Thus, k-m=½ not an integer


Prime / CompositePrime / CompositePrime / CompositePrime / Composite

Define predicate prime(n):n is a prime number iffn>1 and for any its positive integer factors r and s one of them is 1nZ [prime(n) n>1 & r,sN [n=rs r=1 v s=1]]

Define predicate composite(n):n is a composite integer iffn is not prime and is greater than 1nZ [composite(n) n>1 & ~prime(n)]


Existential StatementsExistential StatementsExistential StatementsExistential Statements

x P(x)Proofs:

ConstructiveConstruct an example of such x

Non-constructiveBy contradiction

Show that if such x does NOT exist than a contradiction can be derived


ExampleExampleExampleExample

Prove that nN a,b prime(a) & prime(b) & (a+b=n)

Proof:n=210a=113b=97

// Piece of cake…


Universal StatementsUniversal StatementsUniversal StatementsUniversal Statements

x P(x)x [Q(x) R(x)]

Proof techniques:ExhaustionBy contradiction

Assume the statement is not trueArrive at a contradiction

DirectGeneralizing from an arbitrary particular member

Mathematical induction (chapter 4)


Example 1Example 1Example 1Example 1

Exhaustion:Any even number between 4 and 30 can be written as a sum of two primes:

4=2+26=3+38=3+5…30=11+19


Problems?Problems?Problems?Problems?

Works for finite domains only

What if I want to prove that for any integer n the product of n and n+1 is even?

Can I exhaust all integer values of n?


Example 2Example 2Example 2Example 2

Theorem: nZ [ even(n*(n+1)) ]Proof:

Consider a particular but arbitrary chosen integer n n is odd or evenCase 1: n is odd

Then n=2k+1, n+1=2k+2n(n+1) = (2k+1)(2k+2) = 2(2k+1)(k+1) = 2p for some integer pSo n(n+1) is even


Example 2 (cont’d)Example 2 (cont’d)Example 2 (cont’d)Example 2 (cont’d)

Case 2: n is evenThen n=2k, n+1=2k+1n(n+1) = 2k(2k+1) = 2p

for some integer p

So n(n+1) is even

Done!


FallacyFallacyFallacyFallacy

Generalizing from a particular but NOT arbitrarily chosen exampleI.e., using some additional properties of nExample:

“all odd numbers are prime”“Proof”:

Consider odd number 3It is primeThus for any odd n prime(n) holds

Such “proofs” can be given for correct statements as well!


PreventionPreventionPreventionPrevention

Try to stay away from specific instances (e.g., 3)

Make sure that you are not using any additional properties of n considered

Challenge your proofTry to play the devil’s advocate and find holes in it…


Other Common Other Common MistakesMistakes

Other Common Other Common MistakesMistakes

Pages 120-121 in the book

Using the same letter to mean different things

Jumping to a conclusionInsufficient justification

Begging the questionassuming the claim first


Rational NumbersRational NumbersRational NumbersRational Numbers

A real number is rational iff it can be represented as a ratio/quotient/fraction of integers a and b (b0)

rR [rQ a,bZ [r=a/b & b0]]

Notes:a is numeratorb is denominatorAny rational number can be represented in infinitely many waysThe fractional part of any rational number written in any natural radix has a period in it


Rational or not?Rational or not?Rational or not?Rational or not?

-12-12/1

3.14593+1459/10000

0.56895689568956895689…5689/9999

1+1/2+1/4+1/8+…2

00/1


Theorem 1Theorem 1Theorem 1Theorem 1

Any number with a periodic fractional part in a natural radix representation is rationalProof:

Constructive, radix=10, no whole part:x=0.n1…nmn1…nm…

x=0.(n1…nm)

x*10m-x=n1…nm

x=n1…nm/(10m-1)


Theorem 2Theorem 2Theorem 2Theorem 2

Any geometric series:S=q0+q1+q2+q3+…where -1<q<1evaluates to S=1/(1-q)

ProofProof idea (informally)More formal proofDefinitions of limits and partial sums


Z Z Q QZ Z Q Q

Every integer is a rational number

Proof : set the denominator to 1Book : page 127


QQQQ

The set of rational numbers is closed with respect to arithmetic operations +, -, *, /

Partial proofs : textbook pages 121-131

Formal proof


Irrational NumbersIrrational NumbersIrrational NumbersIrrational Numbers

So far all the examples were of rational numbers

How about some irrationals?esqrt(2)

lecture 07© vadim bulitko : cmput 272, winter 2004, uofa1 cmput 272 formal systems logic in cs...

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