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CS2800-Logic_part2_v.4

# CS2800-Logic_part2_v.4 - Discrete Structures CS 2800 Prof...

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1 Discrete Structures CS 2800 Prof. Bart Selman [email protected] Module Logic (part 2 --- proof methods)

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2 Methods for Proving Theorems
3 Theorems, proofs, and Rules of Inference When is a mathematical argument correct? What techniques can we use o construct a mathematical argument? Theorem – statement that can be shown to be true. Axioms or postulates – statements which are given and assumed to be true. Proof – sequence of statements, a valid argument, to show that a theorem is true. Rules of Inference – rules used in a proof to draw conclusions from assertions known to be true. Note: Lemma is a “pre-theorem” or a result that needs to be proved to prove the theorem; A corollary is a “post-theorem”, a result which follows directly the theorem that has been proved. Conjecture is a statement believed to be true but for which there is not a proof yet. If the conjecture is proved true it becomes a thereom. Fermat’s theorem was a conjecture for a long time.

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4 Valid Arguments Recall: An argument is a sequence of propositions . The final proposition is called the conclusion of the argument while the other propositions are called the premises or hypotheses of the argument. An argument is valid whenever the truth of all its premises implies the truth of its conclusion. How to show that q logically follows from the hypotheses (p 1 p 2 p n )? Show that (p 1 p 2 p n ) q is a tautology One can use the rules of inference to show the validity of an argument. Vacuous proof - if one of the premises is false then (p1 p2 pn) q is vacuously True, since False implies anything. (reminder)
5 Note: Many theorems involve statements for universally quantified variables: e.g. “If x>y, where x and y are positive real numbers, then x 2 > y 2 Quite often, when it is clear from the context, theorems are proved without explicitly using the laws of universal instantiation and universal generalization.

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6 Methods of Proof Direct Proof Proof by Contraposition Proof by Contradiction Proof of Equivalences Proof by Cases Existence Proofs Counterexamples
7 Direct Proof Proof of a statement p q Assume p From p derive q.

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8 Example - direct proof Here’s what you know: Theorem: Mary is a Math major or a CS major. If Mary does not like discrete math, she is not a CS major. If Mary likes discrete math, she is smart. Mary is not a math major. Can you conclude Mary is smart? M C ¬ D ¬ C D S ¬ M ((M C) ( ¬ D ¬ C) (D S) ( ¬ M)) S ? Let M - Mary is a Math major C – Mary is a CS major D – Mary likes discrete math S – Mary is smart Informally, what’s the chain of reasoning?
9 Example - direct proof In general, to prove p q, assume p and show that q follows.

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