# Proof techniques - PowerPoint PPT Presentation

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Proof techniques. CHAPTER TWO. Formal Logic vs. Real-world Arguments. Real-world arguments, unlike the formal proofs of Chapter 1, are normally dependent on context, not the structure of the argument.

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Proof techniques

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## Proof techniques

CHAPTER TWO

### Formal Logic vs. Real-world Arguments

• Real-world arguments, unlike the formal proofs of Chapter 1, are normally dependent on context, not the structure of the argument.

• In other words, a real-world argument may not be universally valid, though it be valid in some important context.

• Terminology: What is a conjecture?

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### Argument Context

• In real-world situations, we often are interested only in the truth of an argument in a particular context.

• Example:“If Mary Beth (or some other student) makes an A in CSC 333, then she must be a bright, hard worker.”

• Call this the Mary Beth Theorem.

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### Examination of the Mary Beth Theorem

• Can we state the M.B. Theorem formally?

• Yes. Let “Mary Beth makes an A in CSC 333” be proposition P, and “Mary Beth is a bright, hard worker” be Q.

• We can state the M.B. Theorem as P -> Q.

• (Or perhaps more properly, let R be “Mary Beth is bright”, and let S be “Mary Beth is a hard worker”; thus, Q can be decomposed as R ^ S

• So, we can state the M.B. Theorem as P-> (R ^ S).

• We can easily establish a truth table for this.

• As stated in the text, if we can’t translate a real-world argument into a formal proof, we should look askance at the argument.

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### Attacking the M.B. Theorem

• Disproving by counterexample:

• Assume that Mary Beth can be shown to be an imbecile, although she has an A in CSC 333.

• This would be a case where R (Mary Beth is bright) is false, making (R ^ S) false.

• So, in at least one case P does NOT imply Q.

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### Which is easier?

• Proving a conjecture using a formal proof, i.e., showing that for all truth values of the propositions, the theorem holds.OR

• Disproving a conjecture by showing one instance in which the theorem “folds” (does not hold), i.e., a counterexample.

• Note that showing one example in which it holds is insufficient as a proof.

• Aside: How does this apply to software testing?

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### Exhaustive Proofs

• If we have a finite population to which we are applying the M.B. Theorem,

• say, the students in CSC 333 in spring of 2010,

• And we can show the truth of the M.B. Theorem for all those students,

• then we have proved the M.B. Theorem by exhaustion.

• Aside: How does this apply to software testing?

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### Direct Proof

• If we want to prove the M.B. Theorem for all students who ever enroll in CSC 333, we might attempt a direct proof:

• Assume P is true.

• Show that the conjecture is universally true because R ^ S inevitably follows from P.

• For the M.B. Theorem, we can’t show this.

• See text for an example of such a proof (p. 92).

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### Contraposition

• Proving by contraposition:

• We already know that P -> Q is logically equivalent to ~Q -> ~P.

• So, if we prove, for example, that if Mary Beth is NOT both bright [R] and a hard worker [S], then Mary Beth will NOT get an A in CSC 333, we have proved the original conjecture.

• Note that the if the contrapositive can be shown to be false in at least one case, this disproves the original conjecture.

• What if Mary Beth is lazy but cheats cleverly?

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### Contradiction

• Proving by contradiction:

• Show that P is true and Q is false is a contradiction, i.e., it is always false.

• In other words, show that it cannot ever be true that P is true and Q is false.

• See example 10, p. 95.

• Proving that something is not true is usually more difficult than assuming it is true and then showing a contradiction.

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### Terms

• Conjecture

• Inductive reasoning

• Deductive reasoning

• Counterexample

• Direct proof

• Contraposition

• Contradiction

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