Proof techniques

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# Proof techniques - PowerPoint PPT Presentation

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

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?

CSC 333

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.

CSC 333

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.

CSC 333

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.

CSC 333

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?

CSC 333

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?

CSC 333

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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• 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.

CSC 333

Terms
• Conjecture
• Inductive reasoning
• Deductive reasoning
• Counterexample
• Direct proof
• Contraposition