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The Foundations: Logic and Proofs

The Foundations: Logic and Proofs. CSC-2259 Discrete Structures. Propositional Logic. Proposition is a declarative statement that is either true of false. Baton Rouge is the capital of Louisiana True Toronto is the capital of Canada False 1+1=2 True 2+2=3 False.

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The Foundations: Logic and Proofs

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  1. The Foundations: Logic and Proofs CSC-2259 Discrete Structures K. Busch - LSU

  2. Propositional Logic Proposition is a declarative statement that is either true of false • Baton Rouge is the capital of Louisiana True • Toronto is the capital of Canada False • 1+1=2 True • 2+2=3 False Statements which are not propositions: • What time is it? • x+1 = 2 K. Busch - LSU

  3. Negation: truth table K. Busch - LSU

  4. Conjunction: truth table K. Busch - LSU

  5. Disjunction: truth table K. Busch - LSU

  6. Exclusive-or: one or the other but not both truth table K. Busch - LSU

  7. (hypothesis) (conclusion) Conditional statement: if p then q p implies q q follows from p p only if q p is sufficient for q truth table K. Busch - LSU

  8. Conditional statement: equivalent (same truth table) Contrapositive: Converse: equivalent Inverse: K. Busch - LSU

  9. Biconditional statement: p if and only if q p iff q If p then q and conversely p is necessary and sufficient for q truth table K. Busch - LSU

  10. Compound propositions Precedence of operators higher lower K. Busch - LSU

  11. Translating English into propositions K. Busch - LSU

  12. Logic and Bit Operations Boolean variables OR AND XOR Bit string K. Busch - LSU

  13. Propositional Equivalences Compound proposition Tautology: always true Contradiction: always false tautology contradiction Contingency: not a tautology and not a contradiction K. Busch - LSU

  14. Logically equivalent compound propositions: is a tautology Have same truth table Example: K. Busch - LSU

  15. De Morgan’s laws K. Busch - LSU

  16. Identity laws Domination laws Idempotent laws Negation laws Double Negation law K. Busch - LSU

  17. Commutative laws Associative laws Distributive laws Absorption laws K. Busch - LSU

  18. Conditional Statements Biconditional Statements K. Busch - LSU

  19. Construct new logical equivalences (De Morgan’s laws) (Double negation law) K. Busch - LSU

  20. Predicates and Quantifiers variable predicate Propositional functions K. Busch - LSU

  21. Predicate logic K. Busch - LSU

  22. Predicate logic predicate 2-ary predicate 3-ary predicate n-ary predicate K. Busch - LSU

  23. Universal quantifier: for all it holds (for every element in domain) is true for every real number (for every element in domain) is not true for every real number Counterexample: K. Busch - LSU

  24. Existential quantifier: there is such that is true because is not true K. Busch - LSU

  25. For finite domain K. Busch - LSU

  26. Quantifiers with restricted domain Precedence of operators higher lower K. Busch - LSU

  27. Bound variable free variable Scope of Scope of K. Busch - LSU

  28. Logical equivalences with quantifiers False False K. Busch - LSU

  29. De Morgan’s Laws for Quantifiers K. Busch - LSU

  30. Example Recall that: K. Busch - LSU

  31. Translating English into Logical Expressions “All hummingbirds are richly colored” “No large birds live on honey” “Birds that do not live on honey are dull in color” “Hummingbirds are small” K. Busch - LSU

  32. Nested Quantifiers Additive inverse Commutative law for addition Associative law for addition K. Busch - LSU

  33. Order of quantifiers K. Busch - LSU

  34. We cannot always change the order of quantifiers true false But not the converse K. Busch - LSU

  35. Translating Math Statements “The sum of two positive integers is always positive” “Every real number except zero has a multiplicative inverse” K. Busch - LSU

  36. For every real number there exists a real number such that whenever K. Busch - LSU

  37. Translating into English “Every student has a computer or has a friend who has a computer” K. Busch - LSU

  38. “There is a student none of whose friends are also friends with each other” K. Busch - LSU

  39. Translating English into Logical Expressions “If a person is female and is a parent, then this person is someone’s mother” female parent mother of K. Busch - LSU

  40. “Everyone has exactly one best friend” Best friends K. Busch - LSU

  41. Negating nested quantifiers Recall: K. Busch - LSU

  42. Rules of Inference If you have a current password, then you can log onto the network You have a current password Therefore, you can log onto the network Modus Ponens Valid argument: if premises are true then conclusion is true K. Busch - LSU

  43. If , then We know that (true) Therefore, (true) Valid argument, true conclusion K. Busch - LSU

  44. If , then We know that (false) Therefore, (false) Valid argument, false conclusion K. Busch - LSU

  45. Modus Ponens If and then K. Busch - LSU

  46. Rules of Inference Modus Ponens Modus Tollens Hypothetical Syllogism Disjunctive Syllogism K. Busch - LSU

  47. Rules of Inference Addition Simplification Conjunction Resolution K. Busch - LSU

  48. It is below freezing now Therefore, it is either below freezing or raining now Addition K. Busch - LSU

  49. It is below freezing and raining now Therefore, it is below freezing now Simplification K. Busch - LSU

  50. If it rains today then we will not have a barbecue today If we do not have a barbecue today then we will have a barbecue tomorrow Therefore, if it rains today then we will have a barbecue tomorrow Hypothetical Syllogism K. Busch - LSU

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