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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

§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

§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

CSCI 115 §2.2 Conditional Statements

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

§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))

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

CSCI 115 §2.3 Methods of Proof

§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?

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

§2.3 – Methods of Proof Valid rules of inference –((p  q)  (q  r))  (p  r) –((p  q)  p)  q Modus Ponens –((p  q)  ~q)  ~pModus Tollens –~~p  pNegation –p  ~~pNegation –p  pRepitition Common mistakes – the following are NOT VALID –((p  q)  q)  p –((p  q)  ~p)  ~q

CSCI 115 §2.4 Mathematical Induction

§2.4 – Mathematical Induction If we want to show P(n) is true  n  Z, n > n 0 where n 0 is a fixed integer, we can do this by: i) Show P(n 0 ) is true Basic step ii) Show that for k > n 0, if P(k) is true, then P(k + 1) is true Inductive step

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