Copyright © Zeph Grunschlag, 2001-2002.

Proofs

Zeph Grunschlag

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Agenda

Proofs: General TechniquesDirect ProofIndirect ProofProof by Contradiction

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Proof Nuts and Bolts

A proof is a logically structured argument which demonstrates that a certain proposition is true. When the proof is complete, the resulting proposition becomes a theorem, or if it is rather simple, a lemma.

For example, consider the proposition:If k is any integer such that k 1 (mod 3),

then k 3 1 (mod 9).Let’s prove this fact:

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ProofsExample

PROVE: kZ k 1(mod 3) k 31(mod 9)

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ProofsExample

PROVE: kZ k 1(mod 3) k 31(mod 9)

1. k 1(mod 3)

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ProofsExample

PROVE: kZ k 1(mod 3) k 31(mod 9)

1. k 1(mod 3)2. n k-1 = 3n

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ProofsExample

PROVE: kZ k 1(mod 3) k 31(mod 9)

1. k 1(mod 3)2. n k-1 = 3n3. n k = 3n + 1

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ProofsExample

PROVE: kZ k 1(mod 3) k 31(mod 9)

1. k 1(mod 3)2. n k-1 = 3n3. n k = 3n + 14. n k 3 = (3n + 1)3

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ProofsExample

PROVE: kZ k 1(mod 3) k 31(mod 9)

1. k 1(mod 3)2. n k-1 = 3n3. n k = 3n + 14. n k 3 = (3n + 1)3

5. n k 3 = 27n 3 + 27n 2 + 9n + 1

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ProofsExample

PROVE: kZ k 1(mod 3) k 31(mod 9)

1. k 1(mod 3)2. n k-1 = 3n3. n k = 3n + 14. n k 3 = (3n + 1)3

5. n k 3 = 27n 3 + 27n 2 + 9n + 16. n k 3-1 = 27n 3 + 27n 2 + 9n

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ProofsExample

PROVE: kZ k 1(mod 3) k 31(mod 9)

1. k 1(mod 3)2. n k-1 = 3n3. n k = 3n + 14. n k 3 = (3n + 1)3

5. n k 3 = 27n 3 + 27n 2 + 9n + 16. n k 3-1 = 27n 3 + 27n 2 + 9n7. n k 3-1 = (3n 3 + 3n 2 + n)·9

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ProofsExample

PROVE: kZ k 1(mod 3) k 31(mod 9)1. k 1(mod 3)2. n k-1 = 3n3. n k = 3n + 14. n k 3 = (3n + 1)3

5. n k 3 = 27n 3 + 27n 2 + 9n + 16. n k 3-1 = 27n 3 + 27n 2 + 9n7. n k 3-1 = (3n 3 + 3n 2 + n)·98. m k 3-1 = m·9

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ProofsExample

PROVE: kZ k 1(mod 3) k 31(mod 9)1. k 1(mod 3)2. n k-1 = 3n3. n k = 3n + 14. n k 3 = (3n + 1)3

5. n k 3 = 27n 3 + 27n 2 + 9n + 16. n k 3-1 = 27n 3 + 27n 2 + 9n7. n k 3-1 = (3n 3 + 3n 2 + n)·98. m k 3-1 = m·99. k 31(mod 9)

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Direct Proofs

Previous was an example of a direct proof. I.e., to prove that a proposition of the form “k P(k) Q(k)” is true, needed to derive that “Q(k) is true” for any k which satisfied “P(k) is true”.

Three basic steps in a direct proof:

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Direct Proofs

1) Deconstruct AxiomsTake the hypothesis and turn it into a

usable form. Usually this amounts to just applying the definition.

EG: k 1(mod 3) really means 3|(k-1) which actually means n k-1 = 3n

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Direct Proofs2) Mathematical Insights

Use your human intellect and get at “real reason” behind theorem.

EG: looking at what we’re trying to prove, we see that we’d really like to understand k 3. So let’s take the cube of k ! From here, we’ll have to use some algebra to get the formula into a form usable by the final step:

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Direct Proofs3) Reconstruct Conclusion

This is the reverse of step 1. At the end of step 2 we should have a simple form that could be derived by applying the definition of the conclusion.

EG. k 31(mod 9) is readily gotten from m k 3-1 = m·9 since the latter is the

definition of the former.

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Proofs Indirect

In addition to direct proofs, there are two other standard methods for proving

k P(k) Q(k)1. Indirect Proof

For any k assume: Q(k)and derive: P(k)

Uses the contrapositive logical equivalence: P Q Q P

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Proofs Reductio Ad Absurdum

2. Proof by Contradiction (Reductio Ad Absurdum)

For any k assume: P(k) Q(k)and derive: P(k) Q(k)

Uses the logical equivalence:

P Q P Q P Q P Q

(P Q ) (P Q ) (P Q ) (P Q )

(P Q ) (P Q )Intuitively: Assume claim is false (so P must be

true and Q false). Show that assumption was absurd (so P false or Q true) so claim true!

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.2. So k 2 is odd.

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.2. So k 2 is odd.3. n k 2 = 2n + 1

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.2. So k 2 is odd.3. n k 2 = 2n + 14. n k 2 - 1 = 2n

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.2. So k 2 is odd.3. n k 2 = 2n + 14. n k 2 - 1 = 2n5. n (k - 1)(k + 1) = 2n

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.2. So k 2 is odd.3. n k 2 = 2n + 14. n k 2 - 1 = 2n5. n (k - 1)(k + 1) = 2n6. 2 | (k - 1)(k + 1)

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.2. So k 2 is odd.3. n k 2 = 2n + 14. n k 2 - 1 = 2n5. n (k - 1)(k + 1) = 2n6. 2 | (k - 1)(k + 1) 7. 2 | (k - 1) 2 | (k + 1) since 2 is prime

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.2. So k 2 is odd.3. n k 2 = 2n + 14. n k 2 - 1 = 2n5. n (k - 1)(k + 1) = 2n6. 2 | (k - 1)(k + 1) 7. 2 | (k - 1) 2 | (k + 1) since 2 is prime8. a k - 1 = 2a b k+1 = 2b

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.2. So k 2 is odd.3. n k 2 = 2n + 14. n k 2 - 1 = 2n5. n (k - 1)(k + 1) = 2n6. 2 | (k - 1)(k + 1) 7. 2 | (k - 1) 2 | (k + 1) since 2 is prime8. a k - 1 = 2a b k+1 = 2b9. a k = 2a + 1 b k = 2b – 1

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.2. So k 2 is odd.3. n k 2 = 2n + 14. n k 2 - 1 = 2n5. n (k - 1)(k + 1) = 2n6. 2 | (k - 1)(k + 1) 7. 2 | (k - 1) 2 | (k + 1) since 2 is prime8. a k - 1 = 2a b k+1 = 2b9. a k = 2a + 1 b k = 2b – 110.In both cases k is odd

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Indirect Proof Example

PROVE: The square of an even number is even. (Ex. 1.16.b)

1. Suppose k 2 is not even.2. So k 2 is odd.3. n k 2 = 2n + 14. n k 2 - 1 = 2n5. n (k - 1)(k + 1) = 2n6. 2 | (k - 1)(k + 1) 7. 2 | (k - 1) 2 | (k + 1) since 2 is prime8. a k - 1 = 2a b k+1 = 2b9. a k = 2a + 1 b k = 2b – 110. In both cases k is odd11. So k is not even

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Rational Numbersan Easier CharacterizationRecall the set of rational numbers Q =

the set of numbers with decimal expansion which is periodic past some point (I.e. repeatinginginginginging…)

Easier characterizationQ = { p/q | p,q are integers with q

0 }Prove that the sum of any irrational

number with a rational number is irrational:

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Reductio Ad AbsurdumExample –English!

You don’t have to use a sequence of formulas. Usually an English proof is preferable! EG:

Suppose that claim is false. So [x is rational and y irrational] [=P] and [x+y is rational] [= Q]. But y = (x+y ) - x. The difference of rational number is rational since

a/b – c/d = (ad-bc)/bd.Therefore [y must be rational] [implies P ]. This contradicts the hypotheses so the assumption that the claim was false was incorrect and the claim must be true. �

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ProofsDisrefutation

Disproving claims is often much easier than proving them.

Claims are usually of the form k P(k). Thus to disprove, enough to find one k –called a counterexample– which makes P(k) false.

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Disrefutation by Counterexample

Disprove: The product of irrational numbers is irrational.

rational. is 12

12product Their 2.

.irrational are Both .2

1 and 2Let 1.

xy

yx

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Blackboard Exercise for 3.1

Proof that the square root of 2 is irrational