《Discrete Mathematic with Applications》读书笔记三

Chapter3 Elementary Number Theory and Methods of proof

The underlying of this chapter is mainly about the question of how to determine the truth or falsity of a mathematical statement.

3.1 Direct Proof and Counterexample I: Introduction

Definitions

Composite number: an integer exactly divisible by at least one number other than itself or 1

The two definition are negations of each other。

Proving Existential Statements

Constructive Proof of existence. Find a x in the domain to make the predicate true.

A nonconstructive proof of existence

Disproving Universal Statements by Counterexample

To disprove a statement means to show that it is false.(or the negation is true)

It is a sign of intelligence to make generalization. Frequently, after observing a property to hold in a large number of cases, you may guess that it holds in all cases. You may , however, run into difficulty when you try to prove your guess. Perhaps you just have not figured out the key to the proof. But perhaps you guess is false. Consequently, when you are having serious difficulty proving a general statement, you should interrupt your efforts to look for a counterexample.

Proving Universal Statements

The method for Exhaustion

In most cases in mathematics, the method of exhaustion  cannot be used.

Directions for Writing Proofs of Universal Statements

1. Copy the statement of the theorem to be proved on your paper.

2. Clearly mark the begining of your proof with the word Proof.

3. Make your proof self-contained.(自足，完备)

Declare of the variable.

4. Write your proof in complete setences.

5. Give a reason for each assertion you make in your proof.

(such as by hypothesis, by definition of, and by theorem...)

6. Include the "little word" that make the logic of your arguments clear.

(Then ,thus, so, hence or there for or it follows)

(Observe that, or Note that, or But ,or Now)

(define the new variable by Let ...)

Variations among Proof

Common Mistakes

1. Arguing from examples.

2. Using the same letter to mean two different things.

3. Jumping to a conclusion.

4. Begging the question.

5. Misuse of the word if.

Getting Proofs Started

V x , if P(x), then Q(x)

Starting Point: Support x is a [particular but arbitrarily chosen] XXX such that P(x) is true.

Conclusiont to Be Show: Q(x) is also true.

Showing That an Existential Statement Is False

It means to show that the negations of the existential statement(an universal statement) is true

Conjecture, Proof, and Disproof

3.2 Direct Proof and Counter Example II:

More On Generalizing from the Generic Particular

Proving Properties of Rational Numbers

Deriving New Mathematics from Old

In the future, when we ask you to prove something directly from the definitions, we will mean that you should restrict yourself to this approach. However, once a collection of statements has been proved directly from the difinitions, another method of proof becomes possible. The statements in the collection can be used to derive additional results.

A corollary is a statement whose truth can be immediately deduced from a therom that has already been proved.

3.3 Direct Proof and Counterexample III: Divisibility

An alternative way to define a prime number is to say that an integer n > 1 is prime if, and only if, its only positive integer divisors are 1 and itself.

Proving Properties of Divisibility

Counterexamples and Divisibility

The Unique Factorization Theorem

It is useful for proving the case that some integerswith the property of combination(such as that the sum of digits is divisible by 3) then the integer n is divisible by x(such as 3).

3.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem

(the proof will be discussed in section 4.4)

div and mod

Computing the Day of the Week

Representations of Integers

Exp 3.4.4 Consecutive Integers Have Opposite Parity(use the method by divided into cases)

The division into cases in a proof is like the transfer of control for an if-then-else statement in a computer program.

if ... ,control transfers to case 1; if not, control transfers to case 2.

There are times when division into more than two cases is called for.(like the switch...case... statement in computer program)

Very few people, when asked to prove an unfamiliar theorem, immediately write down the kind of formal proof you find in a methematics text. Most need to experiment with several possible approaches before they find on that works. A formal proof is much like the ending of a mystery story-the part in which the action of the story is systematically reviewed and all the loose ends are carefully tied together.

Desperation can spur creativity. When you have tried all the obvious approaches without success and you really care about solving a problem, you reach into the odd corners of your memory for anything that may help.

Note that the result of theorem 3.4.3 can also be written. "For any odd integer n, n^2  mod 8 = 1"

3.5 Direct Proof and Counterexample V: Floor and Ceiling

The concept of floor and ceiling are important in analyzing the efficiency of many computer algorithms.

Note that on some calculators floor of x is denoted INT(x)

Proving a Property of Floor

the above proof has a print error. the last equation  if  the floor of (n+ M) = the floor of (x) + m.

The analysis of a number of computer algorithms, such as the binary search and merge sort algorithms, requires that you know the value of floor of (n/2). where n is an integer. The formula for computing this value depends on whether n is even or odd.

The formula to calculate the week of day for  Jan 1 of the Year N.  Define( INT(x) equal to the floor of x,  % equal to mod def in Discrete Mathematics.

d is the week of day.( Sunday : 0  ,  Monday : 1,  Tuesday : 2,  Wednesday : 3, Thursday : 4,  Friday : 5 , Saturday : 6)

d = (n + INT((n - 1) / 4) - INT((n - 1) / 100) + INT((n - 1) / 400)) % 7

Interpretation: One day is added for every four years, except for that each century the day is not added unless the century is a multiple of 400.

3.6 Indirect Argument: Contradiction and Contraposition

The indirect proof ,argument by contradiction, is based on the fact that either a statement is true or it is false but not both.

Knowns as reductio ad impossible or reductio ad absurdum.

Some guidelines for proof by contradiction: if you want to show that there is no object with a certain property, or if you want to show that a certain object does not have a certain property.

For all real number x, y if x is irrational and y is rational, then x+y is irrational.

Negation:  There exists a irrational number x and a rational number y such that x + y is rational.

Argument by Contraposition

Modus tollens.

The second form of indirect argument ,argument by contraposition.

Relation between Proof by Contradiction and Proof by Contraposition

Using the Proof by contradiction for Proposition 3.6.4

The advantage of contraposition is that you could avoiding the negation of a complicated statement.  and the target is clear (to the negation of hypothesis)

And sometimes the target for the contradiction is difficult to find.

The disadvantage of contraposition is that :The contraposition are useful for universal condition statements. 

Any statement that can be proved by contraposition can be proved by contradiction. But the converse is not true.

Such as the square root of 2 is irration could be proved by contradiction but not by contraposition.

Proof as a Problem-Solving Tool

Guideline:

direct proof, 

disproof by counterexample,

proof by contradiction,

proof by contraposition

The tip for sloving problem from P0 178, Susana, EPP, Discrete Mathematics with its application.

  Sloving problems, especially difficult problems, is rarely a straightforward process. At any stage of following the guidelines above, you might want to try the method of a previous stage again. If ,for example ,you fail to find a counterexample for a certain statement, your experience in trying tro find it might help you decide to reattempt a direct argument rather than trying an indirect one. Psychologists who have studied problem solving have found that the most successful problem solvers are thos who are flexible and willing to use a variety of approaches without getting stuck in any one of them for very long. Mathematicians sometimes work for months(or longer) on difficult problems. Don't be discouraged if some problems in this book take you quite a while to solve.

  Learning  the skill of proof and disproof is much like learning other skills, such as those used in swimming, tennis, or playing a musical instrument. When you first start out, you may feel bewildered by all the rules, and you may not feel confident as you attempt new things. But with practice the fules become internalized and you can use them in conjuction with all your other powers- or balance, coordination, judgment, aesthetic sence- to concentrate on winning a meet, winning a match , or playing a concert successfully.

3.7 Two Classical Theorems

square root of 2 is irrational.

there are infinite many prime numbers.

The Irrationality of square root of 2

The Infinitude of the Set of Prime Numbers

Euclid has proof the theorem that the set of all prime numbers is infinite.

It use the face that. If a prime number divides an integer a, then it does not divide a + 1.

The idea of Euclid's proof is this: Suppose the set of prime numbers were finite.(Suppose not). Then you could take the product of all the prime numbers and add one. By theorem 3.3.2 this number must be divisible by some prime number. But by Proposition 3.7.3, this number is not divisible by any of the prime numbers in the set. Hence there must be a prime number that is not in the set of all prime numbers, which is impossible.

The proof of theorem 3.7.4 show that if you form the product of all prime numbers up to a certain point and add one, the result, N, is divisible by a prime number not on the list. The Proof does not show that N is, itself, prime.

When to Use Indirect Proof

Try first to prove a statement directly. Then, if that does not succeed, look for a counterexample. If the search for a counterexample is unsuccessful, look for a proof by contradiction or contraposition.

Open Questions in Number Theory

Such as Mersenne primes.  (2^p -1 where p is prime, Not all of the number of these form are prime)

Is there are infinitely many Mersenne primes?  (The answer is not known)

Another formula to produce a relatively large number of prime numbers is due to Fermat. 

Fermat primes  2^(2^n) + 1, n is a positive integer. (Infinite? not known)

twin primes conjecture

3.8 Application: Algorithms

The word algorithms refers to a step-by-step method for performaing some action.

An Algorithmic Language

Pseudocode, combining elements of Pascal, C, java, and VB.net, but fairly precise, English

Formal constructs of computer languages- such as assignment statements, loops,(ignore the technical details, such as the requirement for explicit end-of-statement delimiters, the range of integer values available on a particular installation.)

variable (as a memory storage)

data type (indicates the set in which the variable takes it values) integers, real numbers, or character string, or set(0, 1)

assignment statement give a vlue to a variable. It has the form

x:= e,

x is a variable, e is a expression.

Condition statements allow to control the direction of the execution.

a. If (condition)

   then s1

   else s2

or

   if ( condition)

   then s1

condition is a predicate involving algorithm variables,  s1 and s2 are algorithm statements or groups of algorithm statements.

bind a group of statements together into a unit by preceding the group with the worddo and following it with the wordsend do.

Iterative statements are used when a sequence of algorithm statements is to be executed over and over again.

We will use two types of iterative statements: while loops andfor-nextloop.

The discussion of while loop can be summarized in a table, called a trace table. That shows the current values of algorithm variables at various points during execution. The trace table for a while loop generally gives all values immediately following each iteration of the loop.("After the zeroth iteration" means the same as "before the first iteration.")

A Notation for Algorithms

We will express algorithms as subroutines that can be called upon by other algorithms as needed and used to transform a set of input variables with given values in to a set of ouotput variables with specific values.

The output variables and their values are assumed to be returned to the calling algorithm.

The division algorithm

q = a div d and r = a mod d (div and mod are the functions built into a number of computer languages.)

The Euclidean Algorithm

gcd(0,0) were defined to be the largest common factor that divides 0 and 0. The problem is that every positive integer divides 0 and there is no largest integer. So there is no largest common factor!.

Calculate the gcd by the unique factorization theorem, sometimes, it unfeasibly long for very large integers.

The Euclid divised a method for finding greatestr common divisors that is easy to use and is much more efficient thant either factoring the numbers or repeatedly test both numbers for divisiblity by successively large integers.

The Euclidean algorithm is based on the two facts, lemma.

A lemma is a statement that does not have much intrinsic interest but helps prove a theorem.

The proof of the second lemma is based on a clever pattern of argument that is used in many different aresas of mathematics:

To prove that A = B ,prove that A <= B and that B <= A