Proof by Contradiction - Hampden-Sydney...
Transcript of Proof by Contradiction - Hampden-Sydney...
Proof by ContradictionLecture 6
Robb T. Koether
Hampden-Sydney College
Mon, Feb 13, 2017
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 1 / 13
Outline
1 Proof by Contradiction
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 2 / 13
Outline
1 Proof by Contradiction
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 3 / 13
Proof by Contradiction
Every statement must be either true or false.It is impossible to correctly deduce a false statement from a truestatement.A false statement can be deduced only from a false statement.This is the basis of proof by contradiction: If the supposition ¬Pleads to a false statement, then ¬P must be false, and therefore Pmust be true.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 4 / 13
Proof by Contradiction
To prove the statement P ⇒ Q by contradiction, we suppose thatits negation is true and then deduce a false statement from thatnegation.The negation of P ⇒ Q is P ∧ ¬Q.To prove by contradiction that P ⇒ Q, we suppose that P is trueand Q is false and then reach a contradiction (a false statement).
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 5 / 13
Example
Theorem√
2 is irrational.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 6 / 13
Example
TheoremLet x be a real number. If x2 = 2, then x 6= a
b for any choice of integersa and b.
Let P be the statement “x2 = 2.”Let Q be the statement “x 6= a
b for any choice of integers a and b.”Then ¬Q is the statement “x = a
b for some choice of integers aand b.”
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 7 / 13
Proof by Contradiction
Proof.We will use proof by contradiction.
Let be x a real number and suppose that x2 = 2 and x = ab for
some choice of integers a and b.Assume that a and b have no factor in common.Then
bx = a,
b2x2 = a2,
2b2 = a2.
Then a2 is even and it follows that a is even. (Why?)
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 8 / 13
Proof by Contradiction
Proof.We will use proof by contradiction.Let be x a real number and suppose that x2 = 2 and x = a
b forsome choice of integers a and b.
Assume that a and b have no factor in common.Then
bx = a,
b2x2 = a2,
2b2 = a2.
Then a2 is even and it follows that a is even. (Why?)
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 8 / 13
Proof by Contradiction
Proof.We will use proof by contradiction.Let be x a real number and suppose that x2 = 2 and x = a
b forsome choice of integers a and b.Assume that a and b have no factor in common.
Then
bx = a,
b2x2 = a2,
2b2 = a2.
Then a2 is even and it follows that a is even. (Why?)
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 8 / 13
Proof by Contradiction
Proof.We will use proof by contradiction.Let be x a real number and suppose that x2 = 2 and x = a
b forsome choice of integers a and b.Assume that a and b have no factor in common.Then
bx = a,
b2x2 = a2,
2b2 = a2.
Then a2 is even and it follows that a is even. (Why?)
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 8 / 13
Proof by Contradiction
Proof.We will use proof by contradiction.Let be x a real number and suppose that x2 = 2 and x = a
b forsome choice of integers a and b.Assume that a and b have no factor in common.Then
bx = a,
b2x2 = a2,
2b2 = a2.
Then a2 is even and it follows that a is even. (Why?)
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 8 / 13
Proof by Contradiction
Proof.We will use proof by contradiction.Let be x a real number and suppose that x2 = 2 and x = a
b forsome choice of integers a and b.Assume that a and b have no factor in common.Then
bx = a,
b2x2 = a2,
2b2 = a2.
Then a2 is even and it follows that a is even. (Why?)
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 8 / 13
Proof by Contradiction
Proof.We will use proof by contradiction.Let be x a real number and suppose that x2 = 2 and x = a
b forsome choice of integers a and b.Assume that a and b have no factor in common.Then
bx = a,
b2x2 = a2,
2b2 = a2.
Then a2 is even and it follows that a is even. (Why?)
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 8 / 13
Proof by Contradiction
Proof.We will use proof by contradiction.Let be x a real number and suppose that x2 = 2 and x = a
b forsome choice of integers a and b.Assume that a and b have no factor in common.Then
bx = a,
b2x2 = a2,
2b2 = a2.
Then a2 is even and it follows that a is even. (Why?)
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 8 / 13
Proof by Contradiction
Proof.Let a = 2k for some integer k .
Then
2b2 = (2k)2,
2b2 = 4k2,
b2 = 2k2.
By the same reasoning as before, b must be even.But a and b have no factor in common.That is a contradiction.Therefore,
√2 is irrational.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 9 / 13
Proof by Contradiction
Proof.Let a = 2k for some integer k .Then
2b2 = (2k)2,
2b2 = 4k2,
b2 = 2k2.
By the same reasoning as before, b must be even.But a and b have no factor in common.That is a contradiction.Therefore,
√2 is irrational.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 9 / 13
Proof by Contradiction
Proof.Let a = 2k for some integer k .Then
2b2 = (2k)2,
2b2 = 4k2,
b2 = 2k2.
By the same reasoning as before, b must be even.But a and b have no factor in common.That is a contradiction.Therefore,
√2 is irrational.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 9 / 13
Proof by Contradiction
Proof.Let a = 2k for some integer k .Then
2b2 = (2k)2,
2b2 = 4k2,
b2 = 2k2.
By the same reasoning as before, b must be even.But a and b have no factor in common.That is a contradiction.Therefore,
√2 is irrational.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 9 / 13
Proof by Contradiction
Proof.Let a = 2k for some integer k .Then
2b2 = (2k)2,
2b2 = 4k2,
b2 = 2k2.
By the same reasoning as before, b must be even.But a and b have no factor in common.That is a contradiction.Therefore,
√2 is irrational.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 9 / 13
Proof by Contradiction
Proof.Let a = 2k for some integer k .Then
2b2 = (2k)2,
2b2 = 4k2,
b2 = 2k2.
By the same reasoning as before, b must be even.
But a and b have no factor in common.That is a contradiction.Therefore,
√2 is irrational.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 9 / 13
Proof by Contradiction
Proof.Let a = 2k for some integer k .Then
2b2 = (2k)2,
2b2 = 4k2,
b2 = 2k2.
By the same reasoning as before, b must be even.But a and b have no factor in common.
That is a contradiction.Therefore,
√2 is irrational.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 9 / 13
Proof by Contradiction
Proof.Let a = 2k for some integer k .Then
2b2 = (2k)2,
2b2 = 4k2,
b2 = 2k2.
By the same reasoning as before, b must be even.But a and b have no factor in common.That is a contradiction.
Therefore,√
2 is irrational.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 9 / 13
Proof by Contradiction
Proof.Let a = 2k for some integer k .Then
2b2 = (2k)2,
2b2 = 4k2,
b2 = 2k2.
By the same reasoning as before, b must be even.But a and b have no factor in common.That is a contradiction.Therefore,
√2 is irrational.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 9 / 13
TheoremLet a, b, c, and d be integers. If a + b + c + d is odd, then a is odd or bis odd or c is odd or d is odd.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 10 / 13
Contradiction vs. Contraposition
Contradiction vs. ContrapositionSuppose we want to prove by contradiction that A ⇒ B.
We begin by supposing that A ∧ ¬B.From that we derive the contradiction A ∧ ¬A and conclude thatA ⇒ B.How does this differ from proof by contraposition?Which method is to be preferred?
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 11 / 13
Contradiction vs. Contraposition
Contradiction vs. ContrapositionSuppose we want to prove by contradiction that A ⇒ B.We begin by supposing that A ∧ ¬B.
From that we derive the contradiction A ∧ ¬A and conclude thatA ⇒ B.How does this differ from proof by contraposition?Which method is to be preferred?
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 11 / 13
Contradiction vs. Contraposition
Contradiction vs. ContrapositionSuppose we want to prove by contradiction that A ⇒ B.We begin by supposing that A ∧ ¬B.From that we derive the contradiction A ∧ ¬A and conclude thatA ⇒ B.
How does this differ from proof by contraposition?Which method is to be preferred?
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 11 / 13
Contradiction vs. Contraposition
Contradiction vs. ContrapositionSuppose we want to prove by contradiction that A ⇒ B.We begin by supposing that A ∧ ¬B.From that we derive the contradiction A ∧ ¬A and conclude thatA ⇒ B.How does this differ from proof by contraposition?
Which method is to be preferred?
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 11 / 13
Contradiction vs. Contraposition
Contradiction vs. ContrapositionSuppose we want to prove by contradiction that A ⇒ B.We begin by supposing that A ∧ ¬B.From that we derive the contradiction A ∧ ¬A and conclude thatA ⇒ B.How does this differ from proof by contraposition?Which method is to be preferred?
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 11 / 13
Example
TheoremLet a and n be integers. If a > 1 and a | n, then a - n + 1.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 12 / 13
Example
TheoremThere are infinitely many primes.
Robb T. Koether (Hampden-Sydney College) Proof by Contradiction Mon, Feb 13, 2017 13 / 13