DISCRETE MATHEMATICS ILECTURES CHAPTER 4Dr. Adam AnthonySpring 2011

Some material adapted from lecture notes provided by Dr. Chungsim Han and Dr. Sam Lomonaco

Section 4.1 Elementary Number Theory Basic Proof Technique: Direct Proof

Tying It All Together: Self-Quiz What is a valid argument? When is a valid argument correct? When

is it wrong? How could you PROVE that a valid

argument is correct?

What is a Proof? A mathematical proof is a valid

argument for which all premises are formally guaranteed (proven) to be true.

Different argument forms lead to different proof types

Sometimes guaranteeing a premise results in having to stop and prove the premise first!

Proof TerminologyAn axiom is a basic assumption about mathematical structures that needs no proof.We can use a proof to demonstrate that a particular statement is true. A proof consists of a sequence of statements that form an argument.The steps that connect the statements in such a sequence are the rules of inference.Cases of incorrect reasoning are called fallacies.A theorem is a statement that can be shown to be true.

Proof Terminology

A lemma is a simple theorem used as an intermediate result in the proof of another theorem.

Lemmas prove premises to be true!

A corollary is a proposition that follows directly from a theorem that has been proved.

A conjecture is a statement whose truth value is unknown. Once it is proven, it becomes a theorem.

Elementary Number Theory Some knowledge we’ll assume:

Properties of Real Numbers in Appendix A (‘basic algebra’)

All logic material from Chapters 2,3 Properties of equality:

A = A (reflexive) If A = B, then B = A (symmetric) If A = B and B = C then A = C (transitive)

The Integers include …-2, -1, 0, 1, 2, … (No Fractions) Integers are closed under +, -, *

Sum/Difference/Product of any two integers is also an integer Integers are not generally closed under /

3/2 is not an integer

Properties of Integers An integer n is even n = 2k for some

integer k An integer n is odd n = 2k + 1 for

some integer k An integer n is prime positive

integers a and b, if n = ab, then a = 1 or b = 1

An integer n is composite positive integers a and b for which n = ab, and a 1 and b 1

Using the symbol means we can use these properties in either direction!

Exercises 1, 2 If r is any integer (even or odd!), is (2r -1) odd?

If s is any integer, is 2s2 + 6s – even?

Which of the following are prime?2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Can an even number >2 be prime? Is every integer > 1 either prime or composite?

Simplest Proof Type: Constructive Proof of Existence

You know this one already! How do we prove:

x in D, Q(x)?1. Give a straight example

Prove that even integer n such that n can be written in 2 ways as a sum of prime numbers

2. Give a set of instructions to find an example: Prove that integer k, 22r + 18s = 2k

Simple ‘Proof’: Disproof by Counter Example

We know this one too! How do we disprove

x in D, P(x)? 2 identical approaches:

1. Directly find a single item in D for which P(x) is false

2. Negate the phrase and prove the negation (this one is a constructive proof!)

Disproof of a Conditional A special case of disproof comes up

again and again: How do we disprove; How do we disprove

x in D, P(x) Q(x)?

In general, we can skip the above and just move to find an example where P(x) is true, but Q(x) is false.

Exercise 3 Disprove the following statements: 1. integers n, n2 – n + 11 is prime

2. real numbers a and b, if a < b, then a2 < b2

3. All primes are odd4. For any integer n, if n2 = 4 then n = 2

Proving Universal Statements We’ve discussed this before: disproving universal

statements is easy, but proving them true is difficult! In general, how do we prove

x in D, P(x)? Exhaustive search: try everything in D!

What if D is infinite??? Use a Generic Particular

Pick a variable x that has a particular but arbitrarily chosen value from the set D

Show that P(x) is true, without ever knowing what number was picked, based on the known axioms for items in D.

Disproving Existential Statements How do we disprove:

x in D, Q(x)? 2 identical approaches:

1. Show that there isn’t a single item in D for which P(x) is false.

2. Negate the phrase and prove the negation using the previous slide:

x in D, Q(x) x in D, P(x)

Exercise 4 True or False?

There is an even integer n > 2 such that n is prime. Perhaps the negation is easier to prove:

For any even integer n, if n > 2, then n is not prime Proof:

1. Pick any even integer at all, we’ll call it x, and suppose that x > 2

2. By definition of an even integer: x = 2m for some integer m3. m must be greater than 1 since x > 24. By definition of a prime integer, if x = a*b then either a or b

must be equal to 15. But x = 2m and 2 1 and m 16. Therefore x is not prime (no matter what even integer you pick!)7. If the negation is TRUE, then the original statement must be

FALSE!!!

Proofs Using Universal Modus Ponens

x in D, P(x) Q(x)P(c) for a particular c in DQ(c)

Proof: MUST SHOW THAT ALL PREMISES ARE GUARANTEED TO BE TRUE! Then, the conclusion follows as a proof

Proving P(c): easy—it’s either true or false for that one thing

How do we prove x in D, P(x) Q(x) is true?

Direct Proof Many proofs start with or otherwise use

a universally quantified conditional statement: x in D, P(x) Q(x)

One way to show this is true is using a Direct Proof:1. Suppose P(x) is true for some particular

but arbitrarily chosen element in D. 2. Show that Q(x) must be true by using

definitions, previously established results, and rules for logical inference.

Tips on Getting Started with a Proof that P(x) Q(x)

1. Read and re-read the question until you clearly understand what is being asked:

2. Write down the starting point, adapted from the left-hand side of the implication—“Suppose there exists a particular but arbitrarily chosen m for which P(m) is true.”

3. Write down a clear statement for what you need to prove: “Therefore, Q(m) is true”

4. On the side, write down every definition and axiom that you know which you think might be relevant

5. From the starting point, use the definitions and axioms to build up toward the conclusion

Exercise 5 Prove that, for all Integers n, if n is even, then

n2 is even. Starting Point: Suppose n is a [particular but

arbitrarily chosen] integer such that n is even. By definition, n = 2k for some integer k Then n2 = (2k)2 = 4k2 = 2(2k2) using basic algebra. Then n2 = 2m where m = 2k2, and 2k2 is an

integer. [Conclusion to be shown: ] Therefore, by

definition n2 is even

Exercise 6 Prove that for all integers m and n, if m

is even and n is odd, then m + n is odd

General Guidelines for Proof Writing

1. Copy the statement of the theorem to be proved onto the paper

2. Clearly mark the beginning of the proof with the word ‘Proof’.

3. Make your proof self-Contained Explain the meaning of each variable you use

4. Write your proof in complete, grammatically correct sentences, mathematical notation excepted.

“Then M + N = 2r + 2s.”

General Guidelines for Proof Writing

5. Be clear about assumptions or things that haven’t been proved yet.

6. Give a reason for each statement By definition, by hypothesis, by previous

theorem, etc. 7. Use good connecting words to guide

logic Then, thus, so, hence, It follows that,

therefore, etc. 8. DISPLAY equations and inequalities

Put them on a SEPARATE line, centered. White space is your friend, improves

readability

PROOFS IN YOUR JOURNAL DO NOT HAVE TO FOLLOW THESE GUIDELINES!•Journal work may be as messy as you like! •You don’t have to use complete sentences, give reasons, or even be clear about assumptions and such•It’s the copying phase—where you re-write for the homework where you apply these 8 guidelines!

Common Proof Mistake: Arguing from examples

“Proof” that integers n, n2 – n + 11 is prime If n = 6, then 62 – 6 + 11 = 41 which is prime

so this must work Using examples is useful when you are

thinking through the problem (esp. in your journal) to visualize structure

But in general, proofs must work for ALL items in the domain, so picking 1 or 2 is not satisfactory proof.

Common Proof Mistake: Using the same letter for two different values

“Suppose m and n are any odd integers. Then by definition of odd, m = 2k + 1 and n = 2k + 1 for some integer k.”

Why is this wrong?

Common Proof Mistake: Jumping to a Conclusion and Circular Reasoning “Proof” that the sum of two even integers is even:

Suppose m and n are any even integers. By definition of even, m = 2r and n = 2s for some integers r and s. Then m + n = 2r + 2s. So m + n is even. Jumping to a conclusion: the last line does not fit any

definition! “Proof” that the product of two odd integers is odd:

Suppose m and n are any odd integers. When any odd integers are multiplied, their product is odd. Hence mn is odd. See the circular reasoning here?

Section 4.3: Divisibility Definition: If n and d are integers and d 0 then

n is divisible by d if, and only if, n equals d times some integer

Identical phrases: n is a multiple of d, d is a factor of n, d is a divisor of n, d divides n

Notation: d | n is read as ‘d divides n’ and means:

integer k, n = dk OR, n/k is an integer

d ∤ n is read as ‘d does not divide n’ and means: integer k, n dk OR, n/k is not an integer

Exercise 1 Is 18 divisible by 6? Does 3 divide 12? Does 5 | 25? If d is a nonzero integer, does d divide

0? If m and n are integers is 10m + 15n

divisible by 5? Does 6 | 3?

Exercise 2 Prove that for all integers a, b and c, if

a|b and b|c then a|c. [This is the transitive property of divisibility]

Exercise 3 Prove that for all integers a and b if a|b

then a2|b2.

Exercise 4 Disprove the following: For all integers a

and b if a|b and b|a then a = b.

Section 4.4: The Quotient-Remainder Theorem Think of an integer as a set of whole items:

12 = xxxxxxxxxxxx Then saying 12 3 is the same as dividing the

set into groups of size 3: 12 3 = xxx xxx xxx xxx,

4 groups of 3 12 = 4*3

What about 13 3? 13 3 = xxx xxx xxx xxx x,

4 groups of 3, with one left over 13 = 4*3 + 1

The Quotient Remainder Theorem For any integer n and a positive integer

d, there exist unique integers q and r such that:

n = dq + r and 0 r<d Find values for q and r for the given n

and d: n = 26, d = 7 n = -26 and d = 7 n = 27 and d = 30 n = -34 and d = 11

Definitions, notation n div d: the value of q (aka the quotient) when n is divided by d

n mod d: the value of r (aka the remainder) when n is divided by d

Find: 29 div 8 29 mod 8 -29 div 8 -29 mod 8

Applying The Quotient-Remainder Theorem Apply the QR theorem to any integer n with d =

2 Just fill in the blanks with what we know,

leaving the rest: For any integer n and the positive integer 2, there

exist unique integers q and r such that: n = 2q + r and 0 r<2

What can we conclude??? The parity property refers to the fact that any

given integer is either even or odd, and not both.

Dealing With Uncertainty Up to now, if we didn’t know something

for sure then we ignored it But we don’t have to! Take any integer n:

Is it even or odd? If it is even, then we know

certain facts about n If it is odd, then we know

a different set of facts

These two ‘possible scenarios’ are referred to as cases

Division into Cases—Argument Form

p rq rs r…z r(p q s … z)r

Cases where r must be true

Assertion that at least one of the cases MUST have occurred.

Division into Cases—Proof Form Given an item r to prove in which some item

can fall into several categories (e.g., odd/even):1. Assert that there exist a fixed number of potential

cases in the world that can be true: (Case1 Case2 …)

2. Set up the following arguments: 1. Case1 r2. Case2 r3. …

3. Perform a direct proof for each case above4. If you succeed for all cases, then by valid

argument, r must always be true.

Proof Example 1 Prove that the product of any two

consecutive integers is even. Starting Point: Select any integer n. Need to prove: the product of n and the

next consecutive integer (n + 1) is even. What do we know for sure about n? What don’t we know for sure?

Proof Example 1 Prove that the product of any two

consecutive integers is even. Select any integer n. Case 1: n is even.

Case 2: n is odd.

Proof Example 2 Prove that if n is an integer, then n2 n Common set of cases: n = 0, n < 0, n

> 0 ‘Divide and conquer’ approach: break

a big problem into 2 or more smaller problems.

Proof Example 3 Show that |xy| = |x||y| where x

and y are real numbers. With two numbers, what are

all the possible scenarios (cases)?

Absolute Value: |x| = x when x 0|x| = -x when x < 0

Proof Example 4: Triangle Inequality

Show that for all real numbers x and y, |x + y| |x| + |y|

Absolute Value: |x| = x when x 0|x| = -x when x < 0

Section 4.6: Indirect Proof Techniques

Proof by contraposition Proof by contradiction

First attempt Let’s try to use a direct proof for the

following: Prove for all integers n, if n2 is even,

then n is even

The Contrapositive Trick Remember that an implication’s

contrapositive is equivalent to the original statement: p q q p

So for any proof statement, we can replace an implication with its contrapositive and prove that instead!

The proof of the new statement is a necessary and sufficient proof of the original

Second Attempt Use contraposition this time: Prove for all integers n, if n2 is even,

then n is even Contrapositive version: for all

integers n, if n is not even, then n2 is not even If n is odd, then n2 is odd Prove

this!

Contraposition Example 1 Prove that for all integers n, if 3n+2 is

odd, then n is odd.

Contraposition Example 2 For all integers n, if 5 ∤ n2 then 5 ∤ n

Proof by Contradiction An inference rule:

p (p c) What is the intuition here?

To use this to our advantage, we use the ‘cause-effect’ interpretation of : If p were true, then it MUST follow that

some contradiction arises If we can show that, then (p c) is true,

which means p is true

Another Attempt Prove by Contradiction that for all

integers, n, if n2 is even, then n is even1. Negate the claim: integer n, n2 is even

n is odd2. Work toward a contradiction3. If/when you find one, proof completed.

Contradiction Example 1 Prove that there is no greatest real

number.

Contradiction Example 2 For all prime numbers a, b and c, a2 + b2

c2

Proof Techniques Summary The only real “technique” we learned is

the direct proof: apply logic to reach a formal conclusion

Division into cases: break large problem up into smaller, manageable pieces

Proof by contraposition: sometimes it is easier to “flip” the problem first.

Proof by contradiction: intuitively the most versatile, but still results in a special kind of direct proof

Proof Practice 1 Prove that if you pick three socks from a

drawer containing just blue socks and black socks, you must get either a pair of blue socks or a pair of black socks.

Proof Practice 2 Prove that if n = ab, where a and b are

positive integers, then a n or b n

Proof Practice 3 Show that at least four of any 22 days

must fall on the same day of the week.

Proof Practice 4 For all integers m and n, if m+ n is even

then m and n are both even or m and n are both odd.