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# 10 - Theorems Introduction and Proofs Discrete Mathematics...

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Introduction Theorems and Proofs Discrete Mathematics Andrei Bulatov

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Discrete Mathematics – Theorems and Proofs 10-2 Previous Lecture Equivalent predicates Equivalent quantified statements Quantifiers and conjunction/disjunction Quantifiers and logic connectives Equivalence and multiple quantifiers
Discrete Mathematics – Theorems and Proofs 10-3 What is a Theorem? The word `theorem’ is understood in two ways First, a theorem is a mathematical statement of certain importance ``A quadratic equation has at most 2 solutions’’ 0 2 = + + c bx ax ``Every statement is equivalent to a certain CNF’’ Second, a theorem is any statement inferred within an axiomatic theory ``Prove that the computer chip design is correct’’

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Discrete Mathematics – Theorems and Proofs 10-4 Axioms In both cases, to infer a theorem we need to start with something. Such starting point is a collection of axioms Two understandings of axioms: - self evident truth ``Two non-parallel lines intersect’’ ``There is something outside me’’ - statements we assume as true, facts from experiment or observation, something we suggest and want to see implications
Discrete Mathematics – Theorems and Proofs 10-5 Axioms (cntd) Self evident truths are usually not quite truths, so we are left with the second meaning of axioms For any two points there is only one line that goes through them

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Discrete Mathematics – Theorems and Proofs 10-6 Proving Theorems To prove theorems we use rules of inference. Usually implicitly In axiomatic theories it is done explicitly: Specify axioms Specify rules of inference Elementary geometry is an axiomatic theory. Axioms are Euclid’s postulates
Discrete Mathematics – Theorems and Proofs 10-7 Proving Theorems We know rules of inference to reason about propositional statements. What about predicates and quantified statements? The simplest method is the method of exhaustion: To prove that 2200 x P(x), just verify that P(a) is true for all values a from the universe. To prove that 5 x P(x), by checking all the values in the universe find a value a such that P(a) is true ``Every car in lot C is red’’ ``There is a blue car in lot C’’

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Discrete Mathematics – Theorems and Proofs 10-8 Rule of Universal Specification Reconsider the argument Every man is mortal. Socrates is a man.
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10 - Theorems Introduction and Proofs Discrete Mathematics...

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