Discrete Mathematics I Lectures Chapter 3

1. Discrete Mathematics ILectures Chapter 3 Some material adapted from lecture notes provided by Dr. Chungsim Han and Dr. Sam Lomonaco Dr. Adam P. AnthonySpring 2011

2. This Week • Introduction to First Order Logic (Sections 3.1—3.3) • Predicates and Logic Functions • Quantifiers • Basic Logic Using Quantifiers • Implication, negation rules for quantifiers

3. Propositional Functions • Propositional function (open sentence): • Statement involving one or more variables, • e.g.: P(x) = x-3 > 5. • Let us call this propositional function P(x), where P is the predicate and x is the variable. What is the truth value of P(2) ? false What is the truth value of P(8) ? false What is the truth value of P(9) ? true

4. Propositional Functions • Let us consider the propositional function Q(x, y, z) defined as: • Q(x, y, z) = x + y = z. • Here, Q is the predicate and x, y, and z are the variables. What is the truth value of Q(2, 3, 5) ? true What is the truth value of Q(0, 1, 2) ? false What is the truth value of Q(9, -9, 0) ? true

5. Function Domains • Propositional functions are just like mathematical functions, they must have a domain: • Real numbers • Integers • People • Students • Professors • Stock Traders? • Domains are used to clarify the purpose of the predicate • Let x be the set of all Students. Let FT(x) = x is a full time student • Sometimes domains are extremely important, particularly with if-then statements

6. Universal Quantification • Let P(x) be a propositional function. • Universally quantified sentence: • For all x in the universe of discourse P(x) is true. • Using the universal quantifier : • x P(x) “for all x P(x)” or “for every x P(x)” • (Note: x P(x) is either true or false, so it is a proposition, not a propositional function.)

7. Universal Quantification • Example: • S(x): x is a B-W student. • G(x): x is a genius. • What does x (S(x) G(x)) mean ? • “If x is a UMBC student, then x is a genius.” • OR • “All UMBC students are geniuses.”

8. Existential Quantification • Existentially quantified sentence: • There exists an x in the universe of discourse for which P(x) is true. • Using the existential quantifier : • x P(x) “There is an x such that P(x).” • “There is at least one x such that P(x).” • (Note: x P(x) is either true or false, so it is a proposition, not a propositional function.)

9. Existential Quantification • Example: • P(x): x is a B-W professor. • G(x): x is a genius. • What does x (P(x) G(x)) mean ? • “There is an x such that x is a UMBC professor and x is a genius.” OR • “At least one B-W professor is a genius.”

10. Quantifiers, Predicates and Domains • A properly defined quantified statement will have predicates and domains clearly specified • How do we say there is a value for x that makes (5x =3) true? • Let x be the set of all real numbers R • Let P(x) = (5x = 3) • xP(x) • Sometimes, this is more trouble than it’s worth to be this clear so we’ll use shorthand: • x in real numbers such that 5x = 3 • Or, even shorter: x in R, 5x = 3 • Finally, if predicates are used (particularly with implication) but no quantifier is given, then assume  is used: • P(x) → Q(x) ≡∀xP(x) → Q(x)

11. Exercise 2.1.1 • Re-write each statement using  and  (sometimes both!) as appropriate: • There Exists a negative real x such that x2=8 • For every nonzero real a, there is a real b such that ab = 1 • All even integers are positive • Some integers are prime • If n2=4 then n = 2

12. Exercise 2.1.2 • Determine the truth values of the following statements: • For all real numbers x, x2 ≥ 0 • For all real numbers x, x2 > 0 • There is an integer n such that n2 = 4 • There is an integer n such that n2 = 3 • For all integers x, If x = 2 then x2 = 4 • If x2 = 4 then x = 2

13. Truth Values of Quantified Statements • Take the statement: • ∀vertebrates a, Bird(a) → Fly(a) • Is it True? • Disproof by Counter-example • Take the statement: • ∃species s, Pig(s) ∧ Fly(s) • Is it True? How do we disprove this one? • Disproof by exhaustive search • Picking domains carefully here can make search easier • In Reality, ∀ is a generalized version of AND (∧) and ∃ is a generalized version of OR ∨: • To say ∀xP(x) means we are saying P(x) is true for everything in the world at the same time • ∀xP(x) ≡P(x1) ∧ P(x2) ∧… ∧P(xn) • To say that ∃xP(x) means we are saying that P(x) is true for at least one (or more or ALL) thing in the world • ∃xP(x) ≡P(x1) ∨ P(x2) ∨… ∨P(xn)

14. Generalized DeMorgan’s • DeMorgan’s law can apply to longer expressions as long as the connective used is the same throughout: • ¬(p ∧ q ∧r ∧z) ≡¬p ∨ ¬q ∨¬r ∨¬z • Repeatedly apply associative laws to see how this works • So if ∀and ∃ are just short-hand for ∧ and ∨ then what happens if we negate them?

15. Negating Quantified Statements • ∀xP(x) ≡P(x1) ∧ P(x2) ∧… ∧P(xn) • ∃xP(x) ≡P(x1) ∨ P(x2) ∨… ∨P(xn) • ¬(∀xP(x)) ≡¬(P(x1) ∧ P(x2) ∧… ∧P(xn)) ≡¬P(x1) ∨ ¬P(x2) ∨… ∨¬P(xn) ≡∃x ¬P(x) • ¬(∃xP(x)) ≡¬(P(x1) ∨ P(x2) ∨… ∨P(xn)) ≡¬P(x1) ∧ ¬P(x2) ∧… ∧¬P(xn) ≡∀x ¬P(x)

16. Tying It All Together • Things seem strange now…logic functions…predicates…quantifiers… • Everything we learned before today is still applicable: • Theorem 2.1.1 (laws for simplification) • Implication elimination/negation • Converse/contrapositive/inverse • Any other equivalences/tautologies/contradictions • Truth tables can be used, but less frequently at this point

17. Exercise 2.1.3 • Write negations for the following statements: • For all numbers x, x2 > 0 • There is an integer n such that n2 = 3 • All even integers are positive • Some integers are prime

18. Exercise 2.1.3 cont. • For any real x, if x ≥ 0, then x2≥x • For any integer n, if n2=n, then n = • Some dogs go to hell • EVERYBODY fails MTH 161!

19. Exercise 2.1.4 • Rewrite the following using ∀ and ∃, then determine the truth value of each statement (hint: negating the statement can help—HOW?): • All even integers are positive • Some integers are prime • There is a positive real x such that x2 ≥ x3 • For any real x, if x ≥ 1, then x2 ≥ x

20. 2.1.4 continued • For any integer n, if n2 = n, then n = 0 • For any real x, if x2 = -1, then x = -1 • If n2 = 4, then n = 2

21. Major Pitfalls with Conditionals • Remember how we interpret implication • If you can’t prove me wrong, then I’m right • For what things in the world is student(x) → smart(x) true? • Smart students • Anybody who is not a student (vacuously true case) • When is ∀people x, student(x) → smart(x) true? When is it false? • When is ∃person x, student(x) → smart(x) true? When is it false? • If we meant to say, ‘there exists a student who is smart’ how do we fix this?

22. Common Uses of Quantifiers • Universal quantifiers are often used with “implies” to form “rules”: (x) student(x) smart(x) means “All students are smart” • Universal quantification is rarely used to make blanket statements about every individual in the world: (x)student(x)smart(x) means “Everyone in the world is a student and is smart” • Existential quantifiers are usually used with “and” to specify a list of properties about an individual: (x) student(x) smart(x) means “There is a student who is smart”

23. Using Multiple Variables, Quantifiers • We already saw a multivariable predicate: Q(x, y, z) = x + y = z. • We can quantify this as (for example): • ∃real x∃realy∃realz, such that Q(x,y,z) • Read this as: there exist real number values x, y, and z such that the sum of x and y is z • We can also mix-and-match quantifiers, but it’s trickier and in English it can be confusing: • ‘There is a person supervising every detail of the production process’ • Work out on the board

24. Understanding Mixed Quantifiers • Here’s how you could ‘determine’ the truth of the following: • ∀x in D, ∃y in E such that P(x,y) • Have a friend pick anything in D, then you have to find something in E that makes P(x,y) true • If you ever fail, then the statement is false (counterexample). • ∃x in D such that ∀y in E, P(x,y) • You need to pick a ‘trump card’: Pick one item from D such that no matter what someone picks out of E, P(x,y) will be true • Your friend should always fail to prove you wrong

25. Exercise 2.2.1 • Express the following using ∀ and ∃, then evaluate the truth of the expression • For any real x, there is a real y such that x + y = 0 • There is a real x such that for any real y, x ≤ y • For any real x, there is a real y such that y < x

26. Less Mathematical Practice (2.2.2) Every gardener likes the sun. xgardener(x) likes(x,Sun) You can fool some of the people all of the time. xtperson(x) time(t)  can-fool(x,t) You can fool all of the people some of the time. xt (person(x) time(t) can-fool(x,t)) x (person(x) t (time(t) can-fool(x,t)) All purple mushrooms are poisonous. x (mushroom(x) purple(x)) poisonous(x) No purple mushroom is poisonous. xpurple(x) mushroom(x) poisonous(x) x (mushroom(x) purple(x)) poisonous(x)

27. Logic to English Translation (2.2.3) • x person(x)  male(x) v female(x) • x male(x) ^ person(x) • x boy(x) male(x) ^ young(x)

28. Negating Mixed Quantifiers • Easy: just apply the negation rule we learned earlier for quantifiers, moving the negation in bit-by-bit: • ¬(∀x in D, ∃y in E such that P(x,y)) ≡∃x in D, ¬(E y in E such that P(x,y)) ≡∃x in D, ∀y in E such that ¬P(x,y) • Works same for ∃x in D such that ∀y in E, P(x,y) • Work out on board!

29. Exercise 2.2.4 • Negate the following until all negation signs are touching a predicate: • ∀x ∀y, P(x,y) • ∀x∃y, (P(x) ∧ Z(x,y)) • ∃x∀y, (P(x) →R(y))

30. Order Matters (half the time)! • If all your quantifiers are the same, you can put them in any order and the meaning remains: • ∀realsx, ∀ realsy, x + y = y + x ≡∀realsy, ∀ realsx, x + y = y + x • Similar for ∃ • You have to be VERY careful about the order of mixed quantifiers: • What is the difference between: ∀people x, ∃a person y such that loves(x,y) ∃person x such that ∀people y, loves(x,y)

31. Valid Arguments Using Quantifiers • Quantifiers help avoid having to name everything in the domain • But what if we reach a point where we are looking at a particular item? • What can we conclude about that item, if all we have a quantified statements?

32. Universal Instantiation • Rule of Universal Instantiation: • If some property is true of EVERYTHING in a domain, then it is true of any PARTICULAR thing in that domain • x in D, P(x) is TRUE for all things in the domain D • Now, observe an item a from the domain D: • Can we conclude anything? • P(a) has to be true

33. Universal Modus Ponens • x in D, P(x)  Q(x) P(a) is true for a particular a in D Therefore, Q(a) is true • Universal instantiation makes this work. How?

34. Universal Modus Tollens • x in D, P(x)  Q(x) Q(a) for some particular a in D Therefore, P(a) • Same Reasoning about Universal Instantiation here, as well!

35. Universal Modus Ponens or Universal Modus Tollens? • All good cars are expensiveA smarty is not expensiveTherefore, a smarty is not a good car • Any sum of two rational numbers is rationalThe numbers a and b are rationalTherefore, a + b is rational

36. Fill In The Blanks (Modus Ponens or Modus Tollens) • If n is even, then n = 2k for some integer k(4x + 2) is evenTherefore, _________________ • If m is odd, then m = 2k + 1 for some integer kr  2i + 1 for any integer ITherefore, __________________ • n is even if and only if n = 2k for some integer k(m + 1)2 = 2l and l is an integerTherefore, __________________

37. Other Quantified Arguments • All of the arguments we looked at in CH 2 have a quantified version of one form or another • Universal Transitivity: • x P(x)  Q(x) x Q(x)  R(x) x P(x)  R(x) • Invalid arguments can be quantified as well, so be careful! • Don’t forget about Converse, Inverse error

38. Diagrams For Analyzing Arguments • All good cars are expensive A smarty is not an expensive car Therefore, a smarty is not a good car Expensive Cars Expensive Cars Smarty Good Cars

39. Diagram Example 1 • All CS Majors are smartPam is not a CS MajorTherefore, Pam is not smart

40. Diagram Example 2 • If a product of two numbers is 0, then at least one of the numbers is 0.x  0 and y  0Therefore, xy  0

41. Diagram Example 3 • No college cafeteria food is goodNo good food is wastedTherefore, No college cafeteria food is wasted

42. Diagram Example 4 • All teachers occasionally make mistakesNo gods ever make mistakesTherefore, No teachers are gods