1 / 25

Propositional Equivalences

Propositional Equivalences. From Aaron Bloomfield.. Used by Dr. Kotamarti. Tautology and Contradiction. A tautology is a statement that is always true p  ¬p will always be true (Negation Law) A contradiction is a statement that is always false p  ¬p will always be false (Negation Law).

adelio
Download Presentation

Propositional Equivalences

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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Propositional Equivalences From Aaron Bloomfield.. Used by Dr. Kotamarti

  2. Tautology and Contradiction • A tautology is a statement that is always true • p  ¬p will always be true (Negation Law) • A contradiction is a statement that is always false • p  ¬p will always be false (Negation Law)

  3. Logical Equivalence • A logical equivalence means that the two sides always have the same truth values • Symbol is ≡or  (we’ll use ≡)

  4. Logical Equivalences of And p T ≡ p Identity law p F ≡ F Domination law

  5. Logical Equivalences of And p  p ≡ p Idempotent law p  q ≡ q  p Commutative law

  6. Logical Equivalences of And (p  q)  r ≡ p  (q  r) Associative law

  7. Logical Equivalences of Or p  T ≡ T Identity law p  F ≡ pDomination law p  p ≡ p Idempotent law p  q ≡ q  p Commutative law (p  q)  r ≡ p  (q  r) Associative law

  8. Corollary of the Associative Law • (p  q)  r ≡ p  q  r • (p  q)  r ≡ p  q  r • Similar to (3+4)+5 = 3+4+5 • Only works if ALL the operators are the same!

  9. Logical Equivalences of Not ¬(¬p) ≡ p Double negation law p  ¬p ≡ T Negation law p  ¬p ≡ F Negation law

  10. Sidewalk chalk guy • Source: http://www.gprime.net/images/sidewalkchalkguy/

  11. DeMorgan’s Law • Probably the most important logical equivalence • To negate pq (or pq), you “flip” the sign, and negate BOTH p and q • Thus, ¬(p  q) ≡ ¬p  ¬q • Thus, ¬(p  q) ≡ ¬p  ¬q

  12. Yet more equivalences • Distributive: p  (q  r) ≡ (p  q)  (p  r) p  (q  r) ≡ (p  q)  (p  r) • Absorption p  (p  q) ≡ p p  (p  q) ≡ p

  13. How to prove two propositions are equivalent? • Two methods: • Using truth tables • Not good for long formula • In this course, only allowed if specifically stated! • Using the logical equivalences • The preferred method • Example: Rosen question 23, page 35 • Show that:

  14. p q r p→r q →r (p→r)(q →r) pq (pq) →r Using Truth Tables

  15. Using Logical Equivalences Original statement Definition of implication DeMorgan’s Law Associativity of Or Re-arranging Idempotent Law

  16. Quick survey • I understood the logical equivalences on the last slide • Very well • Okay • Not really • Not at all

  17. Logical Thinking • At a trial: • Bill says: “Sue is guilty and Fred is innocent.” • Sue says: “If Bill is guilty, then so is Fred.” • Fred says: “I am innocent, but at least one of the others is guilty.” • Let b = Bill is innocent, f = Fred is innocent, and s = Sue is innocent • Statements are: • ¬s  f • ¬b → ¬f • f  (¬b  ¬s)

  18. b f s ¬b ¬f ¬s ¬sf ¬b→¬f ¬b¬s f(¬b¬s) Can all of their statements be true? • Show: (¬s  f)  (¬b → ¬f)  (f  (¬b  ¬s))

  19. Are all of their statements true?Show values for s, b, and f such that the equation is true Original statement Definition of implication Associativity of AND Re-arranging Idempotent law Re-arranging Absorption law Re-arranging Distributive law Negation law Domination law Associativity of AND

  20. What if it weren’t possible to assign such values to s, b, and f? Original statement Definition of implication ... (same as previous slide) Domination law Re-arranging Negation law Domination law Domination law Contradiction!

  21. Quick survey • I feel I can prove a logical equivalence myself • Absolutely • With a bit more practice • Not really • Not at all

  22. Logic Puzzles • Rosen, page 23, questions 19-23 • Knights always tell the truth, knaves always lie • A says “At least one of us is a knave” and B says nothing • A says “The two of us are both knights” and B says “A is a knave” • A says “I am a knave or B is a knight” and B says nothing • Both A and B say “I am a knight” • A says “We are both knaves” and B says nothing

  23. Sand Castles

  24. Quick survey • I felt I understood the material in this slide set… • Very well • With some review, I’ll be good • Not really • Not at all

  25. Quick survey • The pace of the lecture for this slide set was… • Fast • About right • A little slow • Too slow

More Related