Lecture10 - Discrete Mathematics Theorems and Proofs 10-2...

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Introduction Theorems and Proofs Discrete Mathematics Andrei Bulatov 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 Axioms and theorems Discrete Mathematics – Theorems and Proofs 10-3 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 10-4 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’’ Discrete Mathematics – Theorems and Proofs 10-5 Rule of Universal Specification Reconsider the argument Every man is mortal. Socrates is a man. Socrates is mortal In symbolic form it looks like 2200 x (P(x) Q(x)) P(Socrates) Q(Socrates) where P(x) stands for x is a man, and Q(x) stands for x is mortal Discrete Mathematics – Theorems and Proofs 10-6 Rule of Universal Specification (cntd) If an open statement becomes true for all values of the universe, then it is true for each specific individual value from that universe 2200 x P(x) P(c) Example Premises: 2200 x (P(x) Q(x)), P(Socrates) Step Reason 1. 2200 x (P(x) Q(x)), premise 2. P(Socrates) Q(Socrates), rule of universal specification 3. P(Socrates) premise 4. Q(Socrates) modus ponens
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Discrete Mathematics – Theorems and Proofs 10-7 Rule of Universal Generalization Let us prove a theorem: If 2x – 6 = 0 then x = 3. Proof
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Lecture10 - Discrete Mathematics Theorems and Proofs 10-2...

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