1. Logic Truth Tables, Propositions, Implications

2. Statements Logic is the tool for reasoning about the truth or falsity of statements. –Propositional logic is the study of Boolean functions –Predicate logic is the study of quantified Boolean functions (first order predicate logic)

3. Arithmetic vs. Logic ArithmeticLogic 0false 1true Boolean variablestatement variable form of functionstatement form value of functiontruth value of statement equality of functionequivalence of statements

4. Notation WordSymbol andv orw implies6 equivalent] not~ not5 parentheses( ) used for grouping terms

5. Notation Examples EnglishSymbolic A and BA v B A or BA w B A implies BA 6 B A is equivalent to BA ] B notA~A notA5A

6. Statement Forms (p v q) and (q v p) are different as statement forms. They look different. (p v q) and (q v p) are logically equivalent. They have the same truth table. A statement form that represents the constant 1 function is called a tautology. It is true for all truth values of the statement variables. A statement form that represents the constant 0 function is called a contradiction. It is false for all truth values of the statement variables.

7. Truth Tables - NOT P 5P T F F T

8. Truth Tables - AND PQPvQTTTTFFFTFFFFPQPvQTTTTFFFTFFFF

9. Truth Tables - OR PQPwQTTTTFTFTTFFFPQPwQTTTTFTFTTFFF

10. Truth Tables - EQUIVALENT PQP]QTTTTFFFTFFFTPQP]QTTTTFFFTFFFT

11. Truth Tables - IMPLICATION PQP6QTTTTFFFTTFFTPQP6QTTTTFFFTTFFT

12. Truth Tables - Example P true means rain P falsemeans no rain Qtrue means clouds Qfalsemeans no clouds

13. Truth Tables - Example PQP6Q P6Q rain cloudsrain  cloudsT rainno cloudsrain  no cloudsF no raincloudsno rain  cloudsT no rainno cloudsno rain  no cloudsT

14. Algebraic rules for statement forms Associative rules: (p v q) v r ] p v (q v r) (p w q) w r ] p w (q w r) Distributive rules: p v (q w r) ] (p v q) w (p v r) p w (q v r) ] (p w q) v (p w r) Idempotent rules: p v p ] p p w p ] p

15. Rules (continued) Double Negation: 55p ] p DeMorgan’s Rules: 5(p v q) ] 5p w 5q 5(p w q) ] 5p v 5q Commutative Rules: p v q ] q v p p w q ] q w p

16. Rules (continued) Absorption Rules: p w (p v q) ] p p v (p w q) ] p Bound Rules: p v 0 ] 0 p v 1 ] p p w 0 ] p p w 1 ] 1 Negation Rules: p v 5p ] 0 p w 5p ] 1

17. A Simple Tautology P  Q is the same as 5Q  5P This is the same as asking: P  Q ] 5Q  5P How can we prove this true? By creating a truth table! PQT T F F T F

18. A Simple Tautology (continued) Add appropriate columns PQ5P 5Q TT FF T F FT F T TF F F TT

19. A Simple Tautology (continued) Remember your implication table! PQ5P 5QP  Q TT FFT T F FTF F T TFT F F TTT

20. A Simple Tautology (continued) Remember your implication table! PQ5P 5QP  Q5Q  5P TT FFTT T F FTFF F T TFTT F F TTTT

21. A Simple Tautology (continued) Remember your implication table! PQ5P5QP  Q 5Q  5P P  Q ] 5Q  5P TT FFTTT T F FTFFT F T TFTTT F F TTTTT Since the last column is all true, then the original statement: P  Q ] 5Q  5P is a tautology Note: 5Q  5P is the contrapositive of P  Q

22. Translation of English If P then Q:P  Q P only if Q:5Q  5Por P  Q P if and only if Q:P ] Q also written as P iff Q

23. Translation of English P is sufficient for Q: P  Q P is necessary for Q: 5P  5Qor Q  P P is necessary and sufficient for Q: P ] Q P unless Q: 5Q  Por 5P  Q

24. Predicate Logic Consider the statement: x 2 > 1 Is it true or false? Depends upon the value of x! What values can x take on (what is the domain of x)? Express this as a function:S(x) = x 2 > 1 Suppose the domain is the set of reals. The codomain (range) of S is a set of statements that are either true or false.

25. Example S(0.9) = 0.9 2 > 1is a false statement! S(3.2) = 3.2 2 > 1 is a true statement! The function S is an example of a predicate. A predicate is any function whose codomain is a set of statements that are either true or false.

26. Note The codomain is a set of statements The codomain is not the truth value of the statements The domain of predicate is arbitrary Different predicates can have different domains The truth set of a predicate S with domain D is the set of the x ε D for which S(x) is true: {x ε D | S(x) is true} Or simply:{x | S(x)}

27. Quantifiers The phrase “for all” is called a universal quantifier and is symbolically written as œ The phrase “for some” is called an existential quantifier and is written as › Notations for set of numbers: RealsIntegers RationalsPrimes Naturals (nonnegative integers)

28. Goldbach’s conjecture Every even number greater than or equal to 4 can be written as the sum of two primes Express it as a quantified predicate It is unknown whether or not Goldbach’s conjecture is true. You are only asked to make the assertion Another example: Every sufficiently large odd number is the sum of three primes.

29. Negating Quantifiers Let D be a set and let P(x) be a predicate that is defined for x ε D, then 5(œ(x ε D), P(x)) ] (›(x ε D), 5P(x)) and 5(›(x ε D), P(x)) ] (œ(x ε D), 5P(x))