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# CSCI 115 - PowerPoint PPT Presentation

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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Presentation Transcript

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

• 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)  ~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

### CSCI 115

§2.4

Mathematical Induction

§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