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DISCRETE MATHEMATICS I LECTURES CHAPTER 3 Dr. Adam P. Anthony Spring 2011 Some material adapted from lecture notes provided by Dr. Chungsim Han and Dr. Sam Lomonaco

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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

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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) ? What is the truth value of P(9) ? false true

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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) ? What is the truth value of Q(9, -9, 0) ? false true

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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

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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.)

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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.”

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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.)

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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.”

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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) x P(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) ≡ ∀ x P(x) → Q(x)

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

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

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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 ∀ x P(x) means we are saying P(x) is true for everything in the world at the same time ∀ x P(x) ≡ P(x 1 ) ∧ P(x 2 ) ∧ … ∧ P(x n ) To say that ∃ x P(x) means we are saying that P(x) is true for at least one (or more or ALL) thing in the world ∃ x P(x) ≡ P(x 1 ) ∨ P(x 2 ) ∨ … ∨ P(x n ) In Reality, ∀ is a generalized version of AND ( ∧ ) and ∃ is a generalized version of OR ∨ : To say ∀ x P(x) means we are saying P(x) is true for everything in the world at the same time ∀ x P(x) ≡ P(x 1 ) ∧ P(x 2 ) ∧ … ∧ P(x n ) To say that ∃ x P(x) means we are saying that P(x) is true for at least one (or more or ALL) thing in the world ∃ x P(x) ≡ P(x 1 ) ∨ P(x 2 ) ∨ … ∨ P(x n )

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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?

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Negating Quantified Statements ∀ x P(x) ≡ P(x 1 ) ∧ P(x 2 ) ∧ … ∧ P(x n ) ∃ x P(x) ≡ P(x 1 ) ∨ P(x 2 ) ∨ … ∨ P(x n ) ¬( ∀ x P(x)) ≡ ¬(P(x 1 ) ∧ P(x 2 ) ∧ … ∧ P(x n )) ≡ ¬P(x 1 ) ∨ ¬P(x 2 ) ∨ … ∨ ¬P(x n ) ≡ ∃ x ¬P(x) ¬( ∃ x P(x)) ≡ ¬(P(x 1 ) ∨ P(x 2 ) ∨ … ∨ P(x n )) ≡ ¬P(x 1 ) ∧ ¬P(x 2 ) ∧ … ∧ ¬P(x n ) ≡ ∀ x ¬P(x)

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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

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Exercise 2.1.3 Write negations for the following statements: a) For all numbers x, x 2 > 0 b) There is an integer n such that n 2 = 3 c) All even integers are positive d) Some integers are prime

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Exercise 2.1.3 cont. a) For any real x, if x ≥ 0, then x 2 ≥x b) For any integer n, if n 2 =n, then n = c) Some dogs go to hell d) EVERYBODY fails MTH 161!

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Exercise 2.1.4 Rewrite the following using ∀ and ∃, then determine the truth value of each statement (hint: negating the statement can help—HOW?): a) All even integers are positive b) Some integers are prime c) There is a positive real x such that x 2 ≥ x 3 d) For any real x, if x ≥ 1, then x 2 ≥ x

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2.1.4 continued a) For any integer n, if n 2 = n, then n = 0 b) For any real x, if x 2 = -1, then x = -1 c) If n 2 = 4, then n = 2

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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?

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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”

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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 ∃ real y ∃ real z, 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

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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

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

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Less Mathematical Practice (2.2.2) Every gardener likes the sun. x gardener(x) likes(x,Sun) You can fool some of the people all of the time. x t person(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. x purple(x) mushroom(x) poisonous(x) x (mushroom(x) purple(x)) poisonous(x)

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Logic to English Translation (2.2.3) a) x person(x) male(x) v female(x) b) x male(x) ^ person(x) c) x boy(x) male(x) ^ young(x)

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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!

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

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Order Matters (half the time)! If all your quantifiers are the same, you can put them in any order and the meaning remains: ∀ reals x, ∀ reals y, x + y = y + x ≡ ∀ reals y, ∀ reals x, 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)

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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?

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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

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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?

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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!

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Universal Modus Ponens or Universal Modus Tollens? a) All good cars are expensive A smarty is not expensive Therefore, a smarty is not a good car b) Any sum of two rational numbers is rational The numbers a and b are rational Therefore, a + b is rational

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Fill In The Blanks (Modus Ponens or Modus Tollens) If n is even, then n = 2k for some integer k (4x + 2) is even Therefore, _________________ If m is odd, then m = 2k + 1 for some integer k r 2i + 1 for any integer I Therefore, __________________ n is even if and only if n = 2k for some integer k (m + 1) 2 = 2l and l is an integer Therefore, __________________

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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

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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 Smarty Expensive Cars Good Cars

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Diagram Example 1 All CS Majors are smart Pam is not a CS Major Therefore, Pam is not smart

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

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Diagram Example 3 No college cafeteria food is good No good food is wasted Therefore, No college cafeteria food is wasted

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Diagram Example 4 All teachers occasionally make mistakes No gods ever make mistakes Therefore, No teachers are gods

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