Questions about Prime Numbers Proving Existential Statements Is 1 - - PowerPoint PPT Presentation

CSE 20—Discrete Math

Summer, 2006 July 12 (Day 3) Number Theory Methods of Proof Instructor: Neil Rhodes Even/Odd Numbers

An integer n is even iff n equals twice some integer. An integer n is odd iff n equals twice some integer plus 1.

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Questions about Even/Odd Numbers

Is 0 even? Is -33 odd? If a and b are integers, is 6a•2b even?

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Prime Numbers

A prime, p, is a positive integer (greater than 1) whose only positive divisors are 1 and p

Quantified statement:

A positive integer which has a positive divisor not equal one is called composite

Quantified statement:

The set of primes is infinite

∀x∈Z+, ∃y∈Z+: y > x P(y)

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Questions about Prime Numbers

Is 1 prime? Is every integer either prime or composite? Is there a range of integers that are either prime or composite? Is the set of primes infinite?

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Proving Existential Statements

Prove existential statement: ∃x∈D: P(x)

Constructive proof

– Display an x – Give a set of directions for finding x

Nonconstructive proof

– Proof by contradiction (assume non-existence and show a contradiction) – Show x must exist

Example:

– In a group of 367 people, at least two share a birthday

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Disproving Universal Statements

Disprove universal statement: ∀x∈D, P(x)

Counterexample

– Show an x in D where not P(x)

Example: All primes are of the form 2n - 1

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Proving Universal Statements

Prove universal statement: ∀x∈D, P(x)Q(X)

Exhaustive enumeration

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Proving Universal Statements

Prove universal statement: ∀x∈D, P(x)Q(x)

Generalizing from the generic particular

– “Suppose x is in D and P(x)” – … – Therefore Q(x)

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Example of Proving Universal Statement

Prove that the square of any odd number is odd

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How to Write a Proof

Copy what is to be proved

The square of any odd number is odd

Start with Proof: The proof should be self-contained

Identify each variable (declare your variables)

– Suppose m is an integer – Let x be a real number greater than 2

Use complete sentences

Don’t just write a sequence of equations Equations are OK, but should be embedded into sentences

Give a reason for assertions you make

By the definition of even, x=2k for some integer k Note that j is an integer since it is the sum of integers multiplied together

Use connective words

Follows from previous thought New thought New Variable

End with what was to be proved

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Proving Existential Statements

Prove existential statement: ∃x∈D: P(x)

Constructive proof

– Display an x – Give a set of directions for finding x

Nonconstructive proof

– Proof by contradiction (assume non-existence and show a contradiction) – Show x must exist

Example:

– In a group of 367 people, at least two share a birthday

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Disproving Existential Statements

Disprove existential statement: ∃x∈D: P(x)

Equivalent to:

– Prove – Or, alternatively, – Therefore, best bet is generalizing from the generic particular.

Example: There exists a prime which can be written as the square of an

integer > 1

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How to Make Mistakes in a Proof

Argue from examples Use the same variable to mean two different things Jumping to a conclusion Begging the question Misuse of the word if

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Getting Started in a Proof

Figure out:

What is given What is to be proved

Example

Graphs with each of their vertices of even degree contain an Euler Cycle.

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Ratinoal Numbers

A number r is rational iff r can be written as the quotient of two integers with a non-zero denominator. A real number that is not rational is

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r is rational <-> exist a, b in Z such that r=a/b and a0

Questions about Even/Odd Numbers

Is 0 rational? Is 5.7823 rational? Is 5.626262… rational? Are all integers rational?

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Rationals are Closed under Addition

A set is closed under an operation if:

the operation yields results that are in the original set

If a and b are rationals, a + b is rational

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Rationals and Irrationals

Rationals are closed under addition, subtraction, multiplication. Are they closed under division? Irrationals are not closed under multiplication

Irrational * irrational may equal rational What about irrational * (non-zero) rational?

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Number Theory

The study of the properties of integers

Mathematics is the queen of the sciences and number theory is the queen of

• mathematics. —Gauss

Relationship to Computer Science

Logical thinking Proof for important fundamental CS theorem very related to number-theory

proof

Cryptography

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Prime Numbers

Unique Prime Factorization (the fundamental theorem of arithmetic)

Any integer 2 can be written as the product of a unique set of prime

numbers.

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Show factor tree of 12

Divisibility

For integers n and d, then n is divisible by d iff n=dk for some integer k

d divides n n is a multiple of d d is a divisor of n d is a factor of n d | n

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Questions about Divisibility

Does 3 divide 36? Is 100 a multiple of 4? Does 3|39? Is 99 a factor of -99? Is 3 a factor of 0? Is 3 a divisor of 99?

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Proof with Divisibility

If a and b are integers, does 4 divide 4a-4b?

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Proof with Divisibility

Proving a number is not a divisor

Prove 3 is not a factor of 98

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Proof with Divisibility

Given integers a, b, c, if a|b and b|c, then a|c

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Prime Numbers

Integers m and n are relatively prime if they share no common factors

We write m ⊥ n

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Quotient-Remainder Theorem

Given any integer n and any positive integer d, there exist unique integers q and r such that

n=dq+r and 0r<d q can be calculated with the div operator r can be calculated with the mod operator

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Modulo Arithmetic

If:

x mod m = x’ mod m y mod m = y’ mod m z mod m = z’ mod m

Then

(x + y) mod m = (x’ + y’) mod m (x - y) mod m = (x’ - y’) mod m xy mod m = x’y’ mod m (xy+z) mod m = (x’y’ +z’) mod m

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Casting out 9’s

Given x, how to calculate x mod 9

Take all the digits of x Add them together. If the result is bigger than 9, use this formula recursively

Shortcut:

As you are adding the digits of x, if you ever have an intermediate value 9,

add its two digits together

If you ever find a 9, throw it away (cast it out) Casting out 9’s

532 + 656 95 ____ 1273

Why it works:

10 mod 9 = 1 mod 9 10n mod 9 = 1 mod 9 10na mod 9 = a mod 9 (10na+10n-1b+10n-2c+…+10y+ z) mod 9 = (a + b + c + … + y + z) mod 9

Limitations:

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5723386 x 51553 _________ 295057718458

Casting out 11’s

Given x, how to calculate x mod 11

Starting from the right, alternately add and subtract each digit

Why it works

10 mod 11 = -1 mod 11 100 mod 11 = (10•10) mod 11 ! (-1•-1) mod 11 = 1 mod 11 if n is

– even: 10n mod 11 = 1 mod 11 – odd: 10n mod 11 -1 mod 11

10na mod 11 = a mod 11 (or 10na mod 11 = -a mod 11) (10na+10n-1b+10n-2c+…+10y+ z) mod 11

= (z + -1•y + … + -1•c + 1•b + -1•b) mod 11 = (z - y + … -c +b - a) mod 11

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532 + 656 95 ____ 1823

5723386 x 51553 _________ 259057718458

Application: ISBN Check digit

Check digit for ISBN:

1•first digit + 2•second digit + 3•third digit … + 9•9th’ digit ___________ total mod 11 = check digit

Alternatively:

10•first-digit

+9•second digit +8•third-digit +2•ninth digit +1•check digit ____________ total = 0 mod 11

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