Csci 115
Download
1 / 15

CSCI 115 - PowerPoint PPT Presentation


  • 177 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' CSCI 115' - amy


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Csci 115

CSCI 115

Chapter 2

Logic


Csci 1151

CSCI 115

§2.1

Propositions and Logical Operations


2 1 propositions and log ops
§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 ops1
§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 ops2
§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 1152

CSCI 115

§2.2

Conditional Statements


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 statements1
§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 statements2
§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 1153

CSCI 115

§2.3

Methods of Proof


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 proof1
§2.3 – Methods of Proof

  • Direct Proof

  • Proof by contradiction

  • Mathematical Induction (§2.4)


2 3 methods of proof2
§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 1154

CSCI 115

§2.4

Mathematical Induction


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


ad