ICS-252 Discrete Structure II

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ICS-252 Discrete Structure II

Lecture 6

Assistant Professor Department of Computer Science & Assistant Professor, Department of Computer Science &

Engineering, University of Hail, KSA.

s.hagahmoodi@uoh.edu.sa

Outlines

• Proof Methods

• Proof Strategies

DMA=Discrete Mathematics and its

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Proof Methods

Proof:A proof is a valid argument that establishes the truth of mathematical statement. There are two types of proofs;

Formal Proof: In this type all steps are supplied and rules for each step in the arguments are given. Useful theorems can be long and hard to follow

hard to follow.

Informal Proof:Proof of theorems designed for the human consumption are always informal proofs. More than one rule of inference may be used in each step. Steps may be skipped. Rules of inference are not explicitly stated.

Proof Methods: The following are the proofs methods;

1) Direct Proof:It is a way of showing the truth of a given statement 1) Direct Proof: It is a way of showing the truth of a given statement by a straightforward combination of established facts.

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ICS‐252 Dr. Salah Omer, Assistant Professor, CSSE, University of Hail.

Proof Methods

Example: Sum of two even integers is an even number.

Proof: Let x and y are two even numbers. Since they are even therefore we can write

x= 2a, y = 2b for all integers a and b. x + y = 2a + 2b = 2 (a + b)

From this it is clear that x + y has 2 as a factor and therefore is even.

Hence, sum of two even integers is an even number.

2) Indirect Proof: It is also known as proof by contradiction. It is a form of proof that establishes the proof or validity of a proposition by showing that proposition is being false would imply a contradiction. A proposition must be either

t f l d it f l it h b h i ibl

true or false and its falsity has been shown impossible, then proposition must be true.

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Proof Methods

Example:For all integers n, if 3n + 1 is even, then n is odd. Solution: Suppose the contradiction that n is not odd. It means n is

even.

We can write for all integers n, 3n + 1 is even then n is even. If n is even mean n is multiple of 2, therefore n = 2a, for integers a. Then

3n + 1 = 3(2a) +1 = 6a + 1 ………. (1) 6a is even because 2(3a). But 6a + 1 is odd. Therefore 3n + 1 is odd from equation (1).

By assuming n is even, we shown that 3n + 1 is odd which is an contradiction to our assumption

contradiction to our assumption.

Therefore if n is odd then 3n +1 is even, which is impossible. It follows that the original statement if 3n + 1 is even, then n is

odd is true.

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Proof Methods

3 -1) Exhaustive Proof:It is also known as proof by cases. It is a special type of proof by cases where each case

involves checking a single example.

Example 1: Prove that , if n is a positive integer with n ≤ 4.

n

1 )

3 (

+

3

≥

Solution: Proof by exhaustion need only to verify for n=1,2,3,4

n=1, (n + 1)3 _{= (2)}3 _{= 8 and 3}n _{= 3}1 _{= 3; It follows 8 > 3;}

n = 2, (n + 1)3 _{= (3)}3 _{= 27 and 3}n _{= 3}2 _{= 9; It follows 27 > 9;}

3 ( 1)3 ₍₄₎3 ₆₄ _{d 3} ₃3 _{27 It f ll} ₆₄ ₂₇

n = 3, (n + 1)3 _{= (4)}3 _{= 64 and 3}n₌₃3 _{= 27; It follows 64 > 27;}

n = 4, (n + 1)3 _{= (5)}3 _{=125 and 3}n _{= 3}4 _{= 81; It follows 125>81}

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Proof Methods

3-2) Proof by cases: A proof by cases must cover all cases that arises in a theorem.

Example 3: Prove that if n is an integer then n2≥_n

Solution: We can prove for every integer by considering three cases when n=0 n≥1 and n≤-1

three cases when n=0, n ≥1 and n ≤ -1

Case (i)n=0, because 02_{=0 therefore n}2≥_{n holds.}

Case (ii) n ≥1, multiply both sides of inequality by positive n, we get n. n ≥ n.1, implies n2≥_{n hold for n}≥_1.

Case (iii) n ≤ -1, however n2≥_{0, implies n}2≥_n.

Hence n2≥_{n hold for all inequalities.}

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Proof Methods

4) Existence Proof: It is a theorem with a statement involving the existential qualifier. It has two types;

a) Constructive Existence Proof: It proves the existence of a mathematical object with certain properties by creating or providing a method to create this object

or providing a method to create this object.

Example 10:Show that there is a positive integer that can be written as the sum of cubes of positive integers in two different ways;

Solution:After doing some computation we found that 1729 = 103_{+ 9}3_{= 12}3_{+ 1}3

a) Non-constructive Existence Proof: It proves the

existence of a mathematical object with certain properties, but does not provide a means of constructing an example.

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Proof Methods

Example 11:Show that there exist irrational numbers x and y such that xy _{is rational.}

Solution:We know that is irrational. Consider the number if it is rational , we have two irrational numbers

2 2

2

x = and y = with xy _rational.

On the other hand if is irrational then we can let

x= and y = ,

2

2 2

2

So that xy ₌

We have not found irrational number x and y such that is xy

rational.

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2

2 )

2 (

2 2

=

2. 2

=

2

=

Proof Methods

Rather we have shown that either the pair have the desired property and we do not know which of these pairs work.

Uniqueness Proofs: It has two fundamental properties; a) Existence: We show that an element x with the desired

property exists.

b) Uniqueness: We show that if , then y does not have the desired property.

Equivalently, we can show that if x and y both have the desired property then x = y

x

y

≠

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Proof Methods

Example 13:Show that if a and b are real numbers and Then there is a unique real number r such that ar + b = 0. Solution: First, note that the real number r = -b/a is a solution

of ar + b = 0, It follows that a(-b/a) + b = -b + b = 0.

0 ≠

a

Consequently, a real number r exist for which ar + b = 0. This is the existence part of the theorem.

Second, suppose that s is a real number such that as + b = 0, then ar + b = as + b, It follows that ar = as. Dividing both side by a, which is nonzero, we have r = s. this means that if , then This is the uniqueness

_s

≠

_r

0 ≠

+

b

as

q

part of the theorem.

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r

s

≠

_as

₊

_b

_≠

₀

Proof Strategies

Proof Strategies: Generally, if the statement is a conditional statement, you should first try a direct proof; if this fails, you can try an indirect proof if neither of these approaches works , you might try a proof by contradiction.

Forward and Backward Reasoning:To begin a direct proof of

Forward and Backward Reasoning:To begin a direct proof of a conditional statement, you start with the premises. Using these premises, together with axioms and known theorems, you can construct a proof using a sequence of steps that leads to the conclusion. This type of reasoning is called

forward reasoning (see example in direct proof). But forward reasoning is often difficult to use to prove more forward reasoning is often difficult to use to prove more complicated results. In such cases it is helpful to use backward reasoning.

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Proof Strategies

Example:

When x and y are distinctive positive real numbers. Solution: We will proof by backward.

xy

Looking for Counterexamples: Counterexamples show that certain statements are false. When confronted with a

conjecture, you might first try to prove this conjecture, and if your attempts are unsuccessful, you might try to find a counterexample

counterexample.

Example 17: Show that the statement “ Every positive integer is the sum of the square of three integers” is false by finding a counterexample.

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Proof Strategies

We try successive positive integers as a sum of three squares, we find that

To show that there are not three squares that add up to 7, as the sum of three squares we can use are those not exceeding 7 namely 0 1 and 4 Because no three terms

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exceeding 7, namely 0, 1, and 4. Because no three terms where each term is 0, 1 or 4 add up to 7. it follows that 7 is a counterexample. We conclude that the statement “ Every positive integer is the sum of the square of three integers” is false.

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Proof Strategies

Tilings: In tilings, checkerboard is a rectangle divided into squares of same size by horizontal and vertical lines. The game of checkers is played on a board with 8 rows and 8 columns; this board is called the standard checkerboard shown in the figure

shown in the figure.

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Proof Strategies

Dominoes: A domino is a rectangular piece that is one square by two squares as shown in figure.

We say that a board is tiled by dominoes when all its squares are covered with no overlapping dominoes and no

dominoes overhanging the board We now develop some dominoes overhanging the board. We now develop some results about tiling boards using dominoes.

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Proof Strategies

Example 18: Can we tile the standard checkerboard using dominoes. How many ways we have to fill it?

Solution: There are the following ways to tile the checkerboard. 1) Tile it by placing 32 dominoes horizontally.

2) Tile it by placing 32 dominoes vertically.

3) Tile it by placing some horizontally and some vertically dominoes.

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Home Work

Question 1:Prove that the only consecutive integers not exceeding 100 that are perfect powers are 8 and 9 (An integer is a perfect power if it equals na_{, where a is an}

integer greater than 1).

Question 2:Formulate a conjecture about the decimal digits

Question 2: Formulate a conjecture about the decimal digits that occur as the final digit of the square of an integer and prove your result.

Question 3:Can we tile a board obtained by removing one of the four corner squares of a standard checkerboard?

Question 4:Can we tile the board obtained by deleting the upper left and lower right corner squares of a standard checkerboard.

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Home Work

Question 5:Prove or disprove that you can use dominoes to tile the standard checkerboard with two adjacent corners removed (that is, corners that are not opposite).

Question 6: Prove that you can use dominoes to tile a

rectangular checkerboard with an even number of squares rectangular checkerboard with an even number of squares.

Question 7:Show that by removing two white squares and two black squares from an 8 x 8 checkerboard (colored as in the text) you can make it impossible to tile the remaining squares using dominoes.

Question 8:Prove that there are 100 consecutive positive integers that are not perfect squares. Is your proof constructive or non-constructive?

Question 9:Prove that there exists a pair of consecutive integers such that one of these integers is a perfect square and the other is a perfect cube.

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Thank you for your Attention.

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