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Week 4 - Wednesday
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What did we talk about last time?
Divisibility
Proof by cases
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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 <
8424432925592889329288197322308900672
459420460792433, but not for that number
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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.
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How many miles does the pigeon travel?
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Theorem: for all integers n, 3n2 + n + 14 is
even
How could we prove this using cases?
Be careful with formatting
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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
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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
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Give the floor for each of the following values
 25/4
 0.999
 -2.01
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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?
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Prove or disprove:
 x, y R, x + y = x + y
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Prove or disprove:
 x R, m Z x + m = x + m
Proof by Contradiction
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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
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Theorem: There is no largest integer.
Proof by contradiction: Assume that there
is a largest integer.
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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
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Theorem: x, y  Z+, x2 – y2  1
Proof by contradiction: Assume there is
such a pair of integers
Theorem: 2 is irrational
Proof by contradiction:
3.
4.
5.
6.
Suppose 2 is rational
2 = m/n, where m,n Z, n  0 and m
and n have no common factors
2 = m2/n2
2n2 = m2
2k = m2, k Z
m = 2a, a Z
7.
8.
9.
10.
2n2 = (2a)2 = 4a2
n2 = 2a2
n = 2b, b Z
2|m and 2|n
1.
2.
11.
QED
2 is irrational
1.
2.
3.
4.
5.
6.
Negation of conclusion
Definition of rational
Squaring both sides
Transitivity
Square2 of integer is integer
Even x implies even x
(Proof on p. 202)
7.
Substitution
8.
Transitivity
2
9.
Even x implies even x
10. Conjunction of 6 and 9,
contradiction
11. By contradiction in 10,
supposition is false
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Claim: ∀𝑎, 𝑝 ∈ ℤ 𝑝 is prime ˄ 𝑝 ∣ 𝑎 → 𝑝 ∤ 𝑎 + 1
Proof by contradiction:
1. Suppose ∃𝑎, 𝑝 ∈ ℤ such that
𝑝 is prime ˄ 𝑝 ∣ 𝑎 ˄ 𝑝 ∣ 𝑎 + 1
2. 𝑎 = 𝑝 ∙ 𝑟, 𝑟 ∈ ℤ
3. 𝑎 + 1 = 𝑝 ∙ 𝑠, 𝑠 ∈ ℤ
4. 𝑎 + 1 − 𝑎 = 1
5. 𝑝 ∙ 𝑠 − 𝑝 ∙ 𝑟 = 1
6. 𝑝(𝑠 − 𝑟) = 1
7. 𝑝 ∣ 1
8. 𝑝 ≤ 1
9. 𝑝 > 1
10. Contradiction
11. ∀𝑎, 𝑝 ∈ ℤ 𝑝 is prime ˄ 𝑝 ∣ 𝑎 → 𝑝 ∤
𝑎+1
QED
1. Negation of conclusion
2.
3.
4.
5.
6.
7.
8.
Definition of divides
Definition of divides
Subtraction
Substitution
Distributive law
Definition of divides
Since 1 and -1 are the only integers
that divide 1
9. Definition of prime
10. Statement 8 and statement 9 are
negations of each other
11. By contradiction at statement 10
Theorem: There are an infinite number of primes
Proof by contradiction:
1.
2.
Suppose there is a finite list of all
primes: p1, p2, p3, …, pn
Let N = p1p2p3…pn + 1, N  Z
pk | N where pk is a prime
pk | p1p2p3…pn + 1
p1p2p3…pn = pk(p1p2p3…pk-1pk+1…pn)
p1p2p3…pn = pkP, P  Z
pk | p1p2p3…pn
pk does not divide p1p2p3…pn + 1
pk does and does not divide
p1p2p3…pn + 1
10. There are an infinite number of
primes
3.
4.
5.
6.
7.
8.
9.
QED
1.
2.
Negation of conclusion
Product and sum of integers
is an integer
3.
Theorem 4.3.4, p. 174
4.
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
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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
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Review for Exam 1
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Exam 1 is Monday in class!