Lecture 10 Methods of Proof

Lecture 10Methods of Proof

CSCI – 1900 Mathematics for Computer ScienceFall 2014Bill Pine

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Lecture Introduction

• Reading– Rosen - Section 1.7, 1.8

• The Nature of Proofs• Components of a Proof• Rule of Inference and Tautology• Proving equivalences• modus ponens• Indirect Method• Proof by Contradiction

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Past Experience

• Until now we’ve used the following methods to write proofs:–Direct proofs with generic elements,

definitions, and given facts–Proof by enumeration of cases such as

when we used truth tables

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General Outline of a Proof

• p q denotes "q logically follows from p“• Implication may take the form (p1 p2 p3

… pn) q• q logically follows from p1, p2, p3, …, pn

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General Outline (cont)

The process is generally written as:p1

p2

p3::pn

q

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Components of a Proof

• The pi's are called hypotheses or premises• q is called the conclusion• Proof shows that if all of the pi's are true,

then q has to be true• If result is a tautology, then the implication

p q represents a universally correct method of reasoning and is called a rule of inference

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Example of a Tautology based Proof• If p implies q and q implies r, then p implies r

p qq r

p r• By replacing the bar under q r with the “”, the

proof above becomes ((p q) (q r)) (p r) • The next slide shows that this is a tautology and

therefore is universally valid.

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Tautology Example (cont)

p q r p q q r (p q) (q r)

p r ((p q) (q r)) (p r)

T T T T T T T TT T F T F F F TT F T F T F T TT F F F T F F TF T T T T T T TF T F T F F T TF F T T T T T TF F F T T T T T

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Equivalences

• Some mathematical theorems are equivalences, i.e., p q.– If and only if– Necessary and sufficient

• The proof of such a theorem requires two proofs:– Must prove p q, and – Must prove q p

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Rule of Inference: modus ponens

• modus ponens – the method of assertingpp q q

• Example: – p: It is snowing today– p q: If it is snowing, there is no school– q: There is no school today

• Supported by the tautology (p (p q)) q

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Show modus ponens = Rule of Inference

p q (p q) p (p q) (p (p q)) q

T T T T T

T F F F T

F T T F T

F F T F T

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Summary of Arguments

• An argument is a sequence of statements– All statement except the last are premises– The final statement is the conclusion– Normally place the symbol in front of the

conclusion• To say that an argument is valid means only

that its form is valid– If the premises are all true the conclusion is true

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Summary of Arguments (cont)

• It is possible for a valid argument to lead to a false conclusion

• It is possible for an invalid argument to lead to a true conclusion

• Carefully distinguish between validity of the argument and truth of the conclusion

• A valid argument is one whose form, when supplied with true premises cannot have a false conclusion

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Invalid Conclusions from Invalid Premises

• Just because the format of the argument is valid does not mean that the conclusion is true. A premise may be false. For example:

Star Trek is real If Star Trek is real we can travel faster than light

We can travel faster than light• Argument is valid since it is in modus ponens form• Conclusion is false because a premise is false

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Invalid Conclusion from Wrong Form• Sometimes, an argument that looks like modus ponens is

actually not in the correct form. For example:If I am watching Star Trek, then I am happyI am happy I am watching Star Trek

• Argument is not valid since its form is:p qq

p

Which is not modus ponens

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Invalid Argument (continued)• Truth table shows that this is not a tautology:

p q (p q) (p q) q ((p q) q) p

T T T T T

T F F F T

F T T T F

F F T F T

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Rule of Inference: Indirect Method

• Another method of proof is to use the tautology:(p q) (~q ~p)

• The form of the proof is (modus ponens):

~q~q ~p~p

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Indirect Method Example

• p: My e-mail address is available on a web site• q: I am getting spam• p q: If my e-mail address is available on a web

site, then I am getting spam• ~q ~p: If I am not getting spam, then my e-mail

address must not be available on a web site• This proof says that if I am not getting spam, then

my e-mail address is not on a web site

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Another Indirect Method Example

• Prove that if the square of an integer is odd, then the integer is odd too.

• p: n2 is odd• q: n is odd• ~q ~p: If n is even, then n2 is even.

• If n is even, then there exists an integer m for which n = 2×m. n2 therefore would equal (2×m)2 = 4×m2 which must be even.

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Rule of Inference:modus tollens

• Another method of proof is to use the tautology (p q) (~q) (~p)

• The form of the proof is:p q~q

~p• Also known as proof by contradiction

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Proof by Contradiction (cont)

p q (p q) ~q (p q) ~q ~p (p q) (~q) (~p)

T T T F F F T

T F F T F F T

F T T F F T T

F F T T T T T

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Proof by Contradiction (cont)

• The best application for this is where you cannot possibly go through a large (potentially infinite) number of cases to prove that every one is true

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Key Concepts Summary

• The Nature of Proofs• Components of a Proof• Rule of Inference and Tautology• Proving equivalences• modus ponens• Indirect Method• Proof by Contradiction