CS322Week 4 - Wednesday

Last time

What did we talk about last time? Divisibility Proof by cases

The Law of Small Numbers I have claimed that many things can

be demonstrated for a small set of numbers that are not actually true for all numbers

Example: GCD(x,y) gives the greatest common

divisor of x and y GCD(n17 + 9, (n+1)17 + 9) = 1 for all

n < 8424432925592889329288197322308900672459420460792433, but not for that number

Questions?

Logical warmup Two friends who live 36 miles apart decide

to meet and start riding their bikes towards each other. They plan to meet halfway. Each is riding at 6mph. One of them has a pet carrier pigeon who starts

flying the instant the friends start traveling. The pigeon flies back and forth at 18mph

between the friends until the friends meet. How many miles does the pigeon travel?

Another proof by cases

Theorem: for all integers n, 3n2 + n + 14 is even

How could we prove this using cases?

Be careful with formatting

Floor and Ceiling

More definitions

For any real number x, the floor of x, written x, is defined as follows: x = the unique integer n such that n ≤

x < n + 1

For any real number x, the ceiling of x, written x, is defined as follows: x = the unique integer n such that n –

1 < x ≤ n

Examples Give the floor for each of the following values

25/4 0.999 -2.01

Now, give the ceiling for each of the same values

If there are 4 quarts in a gallon, how many gallon jugs do you need to transport 17 quarts of werewolf blood?

Does this example use floor or ceiling?

Proofs with floor and ceiling Prove or disprove:

x, y R, x + y = x + y

Prove or disprove: x R, m Z x + m = x + m

Student LectureProof by Contradiction

Indirect Proof

Proof by contradiction

The most common form of indirect proof is a proof by contradiction

In such a proof, you begin by assuming the negation of the conclusion

Then, you show that doing so leads to a logical impossibility

Thus, the assumption must be false and the conclusion true

Contradiction formatting A proof by contradiction is different from a direct

proof because you are trying to get to a point where things don't make sense

You should always mark such proofs clearly Start your proof with the words Proof by

contradiction Write Negation of conclusion as the

justification for the negated conclusion Clearly mark the line when you have both p and

~p as a contradiction Finally, state the conclusion with its justification

as the contradiction found before

Example

Theorem: There is no largest integer.

Proof by contradiction: Assume that there is a largest integer.

Another example

Theorem: There is no integer that is both even and odd.

Proof by contradiction: Assume that there is an integer that is both even and odd

Another example

Theorem: x, y Z+, x2 – y2 1Proof by contradiction: Assume

there is such a pair of integers

Two Classic Results

Square root of 2 is irrational

1. Suppose is rational2. = m/n, where m,n Z, n 0 and

m and n have no common factors3. 2 = m2/n2

4. 2n2 = m2

5. 2k = m2, k Z6. m = 2a, a Z

7. 2n2 = (2a)2 = 4a2

8. n2 = 2a2

9. n = 2b, b Z10. 2|m and 2|n

11. is irrational

QED

1. Negation of conclusion2. Definition of rational

3. Squaring both sides4. Transitivity5. Square of integer is integer6. Even x2 implies even x

(Proof on p. 202)7. Substitution8. Transitivity9. Even x2 implies even x10. Conjunction of 6 and 9,

contradiction11. By contradiction in 10,

supposition is false

Theorem: is irrationalProof by contradiction:

2

22

2

Proposition 4.7.3 Claim: Proof by contradiction:1. Suppose such that

2. 3. 4. 15. 1

6. 7. Contradiction

8. Negation of conclusion

9. Definition of divides10.Definition of divides11.Subtraction12.Substitution13.Distributive law14.Definition of divides15.Since 1 and -1 are the only

integers that divide 116.Definition of prime17.Statement 8 and statement 9

are negations of each other18.By contradiction at statement

10QED

Infinitude of primes

1. Suppose there is a finite list of all primes: p1, p2, p3, …, pn

2. Let N = p1p2p3…pn + 1, N Z

3. pk | N where pk is a prime4. pk | p1p2p3…pn + 15. p1p2p3…pn = pk(p1p2p3…pk-1pk+1…pn)6. p1p2p3…pn = pkP, P Z7. pk | p1p2p3…pn

8. pk does not divide p1p2p3…pn + 19. pk does and does not divide p1p2p3…

pn + 110. There are an infinite number of primes

QED

1. Negation of conclusion

2. Product and sum of integers is an integer

3. Theorem 4.3.4, p. 1744. Substitution5. Commutativity6. Product of integers is

integer7. Definition of divides8. Proposition from last slide9. Conjunction of 4 and 8,

contradiction10. By contradiction in 9,

supposition is false

Theorem: There are an infinite number of primesProof by contradiction:

A few notes about indirect proof Don't combine direct proofs and

indirect proofs You're either looking for a

contradiction or not Proving the contrapositive directly is

equivalent to a proof by contradiction

Quiz

Upcoming

Next time…

Review for Exam 1

Reminders

Exam 1 is Monday in class!