Week 4 - Wednesday
What did we talk about last time? Divisibility Proof by cases
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(n , (n+1) ) = 1 for all n < , but not for that number
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?
Theorem: for all integers n, 3n 2 + n + 14 is even How could we prove this using cases? Be careful with formatting
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
Give the floor for each of the following values 25/4 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?
Prove or disprove: x, y R, x + y = x + y Prove or disprove: x R, m Z x + m = x + m
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
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
Theorem: There is no largest integer. Proof by contradiction: Assume that there is a largest integer.
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
Theorem: x, y Z +, x 2 – y 2 1 Proof by contradiction: Assume there is such a pair of integers
1. Suppose is rational 2. = m/n, where m,n Z, n 0 and m and n have no common factors 3. 2 = m 2 /n n 2 = m k = m 2, k Z 6. m = 2a, a Z 7. 2n 2 = (2a) 2 = 4a 2 8. n 2 = 2a 2 9. n = 2b, b Z 10. 2|m and 2|n 11. is irrational QED 1. Negation of conclusion 2. Definition of rational 3. Squaring both sides 4. Transitivity 5. Square of integer is integer 6. Even x 2 implies even x (Proof on p. 202) 7. Substitution 8. Transitivity 9. Even x 2 implies even x 10. Conjunction of 6 and 9, contradiction 11. By contradiction in 10, supposition is false Theorem: is irrational Proof by contradiction:
QED
1. Suppose there is a finite list of all primes: p 1, p 2, p 3, …, p n 2. Let N = p 1 p 2 p 3 …p n + 1, N Z 3. p k | N where p k is a prime 4. p k | p 1 p 2 p 3 …p n p 1 p 2 p 3 …p n = p k (p 1 p 2 p 3 …p k-1 p k+1 …p n ) 6. p 1 p 2 p 3 …p n = p k P, P Z 7. p k | p 1 p 2 p 3 …p n 8. p k does not divide p 1 p 2 p 3 …p n p k does and does not divide p 1 p 2 p 3 …p n 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 Substitution 5. Commutativity 6. Product of integers is integer 7. Definition of divides 8. Proposition from last slide 9. Conjunction of 4 and 8, contradiction 10. By contradiction in 9, supposition is false Theorem: There are an infinite number of primes Proof by contradiction:
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
Review for Exam 1
Exam 1 is Monday in class!