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CSCI 115

CSCI 115. Chapter 2 Logic. CSCI 115. §2 .1 Propositions and Logical Operations. §2 .1 – Propositions and Log Ops. Logical Statement Logical Connectives Propositional variables Conjunction (and: ) Disjunction (or: ) Negation (not: ~) Truth tables.

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CSCI 115

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  1. CSCI 115 Chapter 2 Logic

  2. CSCI 115 §2.1 Propositions and Logical Operations

  3. §2.1 – Propositions and Log Ops • Logical Statement • Logical Connectives • Propositional variables • Conjunction (and: ) • Disjunction (or: ) • Negation (not: ~) • Truth tables

  4. §2.1 – Propositions and Log Ops • Quantifiers • Consider A = {x| P(x)} • t  A if and only if P(t) is true • P(x) – predicate or propositional function • Programming • if, while • Guards

  5. §2.1 – Propositions and Log Ops • Universal Quantification – true for all values of x • x P(x) • Existential Quantification – true for at least one value • x P(x) • Negation of quantification

  6. CSCI 115 §2.2 Conditional Statements

  7. §2.2 – Conditional Statements • Conditional statement: If p then q • p  q • p – antecedent (hypothesis) • q – consequent (conclusion) • Truth tables

  8. §2.2 – Conditional Statements • Given a conditional statement p  q • Converse • Inverse • Contrapositive • Biconditional (if and only if) • p  q is equivalent to ((p  q)  (q  p))

  9. §2.2 – Conditional Statements • Statements • Tautology (always true) • Absurdity (always false) • Contingency (truth value depends on the values of the propositional variables) • Logical equivalence ()

  10. CSCI 115 §2.3 Methods of Proof

  11. §2.3 – Methods of Proof • Prove a statement • Choose a method • Disprove a statement • Find a counterexample • Prove or disprove a statement • Where do I start?

  12. §2.3 – Methods of Proof • Direct Proof • Proof by contradiction • Mathematical Induction (§2.4)

  13. §2.3 – Methods of Proof • Valid rules of inference • ((p  q)  (q  r))  (p  r) • ((p  q)  p)  q Modus Ponens • ((p  q)  ~q)  ~p Modus Tollens • ~~p  p Negation • p  ~~p Negation • p  p Repitition • Common mistakes – the following are NOT VALID • ((p  q)  q)  p • ((p  q)  ~p)  ~q

  14. CSCI 115 §2.4 Mathematical Induction

  15. §2.4 – Mathematical Induction • If we want to show P(n) is true nZ, n > n0 where n0 is a fixed integer, we can do this by: i) Show P(n0) is true • Basic step ii) Show that for k > n0, if P(k) is true, then P(k + 1) is true • Inductive step

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