1. ICS 6D Logic Overview Instructor: Sandy Irani

2. What is logic? Branch of mathematics in which variables and expressions have value true (T) or false (F) Particularly relevant to CS: – AI: automated reasoning – Digital logic design Useful in any area in which it is important to make precise statements and reason in a systematic way.

3. Proposition A proposition is a statement that is either true or false: – 2 + 3 = 5 – 3 + 4 = 6 – 4 is a prime number – It is raining today – It will rain tomorrow (unknown truth value) – Chocolate is the best flavor of ice cream (matter of opinion)

4. Propositions Statements that are not propositios: – How are you doing? (Question) – Have a nice day! (Command) – Bummer! (exclamation) Propositional variables – Typically: p, q, r, s – Have value T or F.

5. Logical Operations: Conjunction Conjunction (AND) – p ˄ q is true when p is true and q is true, false otherwise Truth table shows the truth value of an expression for every possible combination of truth values for the individual propositions In the expression. pq p˄qp˄q

6. Logical Operations: Disjunction Disjunction (OR) – p ˅ q is true when p is true or q is true or both are true. p ˅ q only when p and q are both false. pq p ˅ q

7. Inclusive vs. Exclusive OR Inclusive OR (True if either or both propositions are true) – The patient should not take the medication if she has a history of migraines or she has diabetes. Exclusive OR (True if exactly one of the propositions is true. False if both propositions are true) – I will drive or ride my bike to work today. In logic: “OR” is always the inclusive OR unless explicitly stated otherwise.

8. Logical Operations: Complement Complement (NOT)  p p pp

9. Compound Propositions p  q   r Order of precedence: – Complement – Conjunction – Disjunction Good to use parens: – Except for complement, multiple , multiple 

10. Compound Propositions p: true q: false r: true  p  (q   r)  (p  q   r)

11. Truth table for:  p  (q   r)  p  (q   r) pqr pp rrq   r

12. Logical Equivalence Two compound propositions are logically equivalent if they have the same truth value for every combination of truth values for their individual propositions.  (p  q)   p   q pq  (p  q)  p   q

13. De Morgan’s Law:  (p  q)   p   q p: The applicant is over 18 years old q: the applicant has a valid driver’s license  (p  q): It is not true that the applicant is over 18 years old and has a valid driver’s license.  p   q: The applicant is not over 18 years old or does not have a valid driver’s license.

14. De Morgan’s Law:  (p  q)   p   q p: the patient has a history of migraines q: the patient has diabetes  (p  q): It is not true that the patient has a history of migrains or has diabetes.  p   q: The does not have a history of migraines and does not have diabetes.

15. The conditional operation “Implies” p → q – p is the “hypothesis” and q is the “conclusion” pqp → q “If p, then q.” “q, if p.” “p implies q”

16. The conditional operation h: you mow my lawn c: I pay you $20 h → c: If you mow my lawn, I will pay you $20. You mow my lawn You don’t mow my lawn I pay you $20 I don’t pay you $20

17. The conditional operation s: Suzy studied for her test p: Suzy passed her test p → q   p  q: – If Suzy studied for her test then she passed her test. – Either Suzy did not study for her test or she passed her test.

18. Contrapositive s: Suzy studied for her test p: Suzy passed her test The contrapositive of p → q is  q →  p. p → q   q →  p – If Suzy studied for her test then she passed her test. – If Suzy did not pass her test, then she did not study for it.

19. The bi-conditional operation “If and only if” p ↔ q – p ↔ q is true whenever p and q have the same truth value. pqp ↔ q

20. Convenient logical equivalences p → q   p  q De Morgans Laws:  (p  q)   p   q  (p  q)   p   q Associative Laws: (p  q)  r  p  (q  r ) (p  q)  r  p  (q  r )

21. Disjunction/Conjunction p 1  p 2  p 3  p 4  ….  p n p 1  p 2  p 3  p 4  ….  p n

22. Predicate P(x): “x is prime” is a predicate (not a proposition) – Truth value depends on the value of x. – P(5): “5 is prime.” P(5) is a proposition. – P(6): “6 is prime.” P(6) is a proposition. Variable x in a predicate has a domain – Set of possible values for x. – For “x is prime” a natural domain is the set of positive integers.

23. Predicate Domain: set of students in a class: – {Abigail, Bernice, Charlie, …, Zachary} Predicate T(x): x got an A on the test. – T(Bernice): “Bernice got an A on the test.” – T(Charlie)  T(Zachary): “Charlie and Zachary both got an A on the test.” – T(Zachary) → T(Bernice): “If Zachary got an A on the test then Bernice got an A on the test.” (All propositions.)

24. Existential Quantifier Domain: set of students in a class: – {Abigail, Bernice, Charlie, …, Zachary} Predicate T(x): x got an A on the test. – ∃ x T(x) “There is a student who got an A on the test.” “Some student got and A on the test.” “At least one student got an A on the test.” For finite domains: – ∃ x T(x)  ( T(Abigail)  T(Bernice)  T(Charlie)  …  T(Zachary))

25. Existential Quantifier ∃ x T(x) True if T(n) is true for at least one value for x in the domain. False if T(n) is false for every value of x in the domain. Example: domain is the set of all positive integers. ∃ x (x 2 = x) ∃ x (x + x = x)

26. Universal Quantifier Domain: set of students in a class: – {Abigail, Bernice, Charlie, …, Zachary} Predicate P(x): x passed the test. – ∀ x P(x) “Every student passed the test.” “All students passed the test.” For finite domains: – ∀ x P(x)  ( P(Abigail)  P(Bernice)  P(Charlie)  …  P(Zachary))

27. Universal Quantifier ∀ x T(x) True if T(n) is true for every value for x in the domain. False if T(n) is false for at least one value for x in the domain. (counterexample) Example: domain is the set of all positive integers. ∀ x (x 2 = x) ∀ x (x 2 ≥ x) ∀ x (x 2 > x)

28. Universal Quantifier Examples P(x): x is prime. D(x): x is odd The domain for x is the set of all positive integers ∀ x (P(x) → D(x)) ∀ x (P(x) → (D(x)  (x = 2)) ) ∀ x ((x < 0) → P(x))