Class 4: Logical Formulas

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. cs2102: Discrete Mathematics Class 4: Logical Formulas David Evans University of Virginia
2. Plan Well-Ordering Principle Review Logical Operations Logical Formulas
3. Odd Summation. Prove that for all n > 0, the sum of the first n odd numbers is n2.
4. Odd Summation. Prove that for all n > 0, the sum of the first n odd numbers is n2. (from Class 3)
5. Odd Summation. Prove that for all n > 0, the sum of the first n odd numbers is n2.
6. Odd Summation. Prove that for all n > 0, the sum of the first n odd numbers is n2.
7. Well-Ordering Principle Proofs Counter-examples to P
8. Logical Operations
9. One-Input Boolean Operations P NOT(P) T F F T Are there any others?
10. One-Input Boolean Operations P IDENT(P) T T F F Are there any others?
11. One-Input Boolean Operations P NOT(P) IDENT(P) TAUT(P) FALSE(P) T F T T F F T F T F How many do we need to produce all of them?
12. P NOT(P) IDENT(P) TAUT(P) FALSE(P) T F T T F F T F T F
13. P NOT(P) IDENT(P) TAUT(P) FALSE(P) T F T T F F T F T F Can we compute anything interesting with one-input Boolean operators?
14. Two-Input Operators
15. Two-Input Operators P Q OP(P, Q) T T T F F T F F
16. Two-Input Operators P Q AND(P, Q) OR(P, Q) P IMPLIES Q P IFF Q XOR(P, Q) T T T T T F F T F T F T F F F F
17. Two-Input Operators P Q AND(P, Q) OR(P, Q) P IMPLIES Q P IFF Q XOR(P, Q) T T T T T T F T F F T F F T F T F T T F T F F F F T T F
18. Implies and Causation What is the relationship between P IMPLIES Q and P “CAUSES” Q?
19. Two-Input Operators P Q AND(P, Q) OR(P, Q) P IMPLIES Q P IFF Q XOR(P, Q) T T T T T T F T F F T F F T F T F T T F T F F F F T T F How many two-input Boolean operators are there?
20. P Q AND IFF XOR OR IMPLIES T T T T F F F T T T F F F T T T F F T F T F T F F T F F T T F T T F T F F T T F F T F F T F T F T T F T T F F F T F F F T F F T F T T F T T T F How many of these do we actually need?
21. DeMorgan’s Laws Augustus De Morgan (1806-1871) We can replace OR with AND, and replace AND with OR!
22. Proof of equivalence:
23. Proof of equivalence:
24. Slack “quiz”: who of these knew De Morgan’s laws? George Boole Ada, Countess of Lovelace Thomas Jefferson Grace Hopper William of Ockham
25. Slack “quiz”: who of these knew De Morgan’s laws? [Formal Logic, 1847] George Boole (1815-1864) Ada Lovelace (1815-1852) Thomas Jefferson (1743-1826) Grace Hopper (1906-1992) William of Ockham (1285-1346)
26. An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities (1854)
28. Completeness of Two-Input Operators Is there something special about AND and OR? Can we find an equivalence to AND that only uses NOT and XOR?
30. P Q AND XOR T T T F T F F T F T F T F F F F
31. P Q AND “FFTF” T T T F T F F F F T F T F F F F
32. P Q AND “FFTF” IFF XOR OR IMPLIES T T T T F F F T T T F F F T T T F F T F T F T F F T F F T T F T T F T F F T T F F T F F T F T F T T F T T F F F T F F F T F F T F T T F T T T F
33. Definitions A formula is valid if there is no way to make it false. A formula is satisfiable if there is some way to make it true.
34. A formula is valid if there is no way to make it false. A formula is satisfiable if there is some way to make it true.
35. A formula is valid if there is no way to make it false. A formula is satisfiable if there is some way to make it true.
36. A formula is valid if there is no way to make it false. A formula is satisfiable if there is some way to make it true.
37. Charge • If you need to enroll in the class, bring me a course action form to sign • Due Friday (6:29pm): PS1 • Finish reading MCS Ch 3 next week