Lecture 1.2: Equivalences, and Predicate Logic*

Lecture 1.2: Equivalences, and Predicate Logic* CS 250, Discrete Structures, Fall 2011 Nitesh Saxena *Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag

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Outline • Equivalences (contd.) • Predicate Logic (Predicates and Quantifiers) Lecture 1.2 - Equivalence, and Predicate Logic

Propositional Logic – Logical Equivalence • p is logically equivalent to q if their truth tables are the same. We write p q. • In other words, p is logically equivalent to q if p  q is True. • There are some famous laws of equivalence, which we review next. Very useful in simplifying complex composite propositions Lecture 1.2 - Equivalence, and Predicate Logic

Identity laws Like adding 0 Domination laws Like multiplying by 0 Idempotent laws Delete redundancies Double negation “I don’t like you, not” Commutativity Like “x+y = y+x” Associativity Like “(x+y)+z = y+(x+z)” Distributivity Like “(x+y)z = xz+yz” De Morgan Laws of Logical Equivalences Rosen; page 27: Table 6 Lecture 1.2 - Equivalence, and Predicate Logic

De Morgan’s Law - generalized De Morgan’s law allow for simplification of negations of complex expressions • Conjunctional negation: (p1p2…pn)  (p1p2…pn) “It’s not the case that all are true iff one is false.” • Disjunctional negation: (p1p2…pn)  (p1p2…pn) “It’s not the case that one is true iff all are false.” • Let us do a quick (white board) proof to show that the law holds Lecture 1.2 - Equivalence, and Predicate Logic

if NOT (blue AND NOT red) OR red then… Another equivalence example (pq)  q p q (pq)  q DeMorgan’s Double negation (p q)  q p (q  q) Associativity p q Idempotent Lecture 1.2 - Equivalence, and Predicate Logic

[p (p  q)]  q  [p (pq)] q  [(pp) (pq)]q  [ F  (pq)]  q  (pq)  q  (pq)  q  (pq)  q  p (q  q )  p T  T Yet another example • Show that[p (pq)]  q is a tautology. • We use to show that [p (pq)]  q  T. substitution for  distributive uniqueness identity substitution for  De Morgan’s associative uniqueness domination Lecture 1.2 - Equivalence, and Predicate Logic

YES NO YES YES Predicate Logic – why do we need it? Proposition, YES or NO? 3 + 2 = 5 X + 2 = 5 X + 2 <= 5 for any choice of X in {1, 2, 3} X + 2 = 5 for some X in {1, 2, 3} Propositional logic can not handle statements such as the last two. Predicate logic can. Lecture 1.2 - Equivalence, and Predicate Logic

Predicate Logic Example Alicia eats pizza at least once a week. Garrett eats pizza at least once a week. Allison eats pizza at least once a week. Gregg eats pizza at least once a week. Ryan eats pizza at least once a week. Meera eats pizza at least once a week. Ariel eats pizza at least once a week. … Lecture 1.2 - Equivalence, and Predicate Logic

Predicates -- Definition Alicia eats pizza at least once a week. Define: P(x) = “x eats pizza at least once a week.” Universe of Discourse - x is a student in cs250 A predicate, or propositional function, is a function that takes some variable(s) as arguments and returns True or False. Note that P(x) is not a proposition, P(Ariel) is. … Lecture 1.2 - Equivalence, and Predicate Logic

NO YES NO NO YES YES Predicates with multiple variables Suppose Q(x,y) = “x > y” Proposition, YES or NO? Q(x,y) Q(3,4) Q(x,9) Predicate, YES or NO? Q(x,y) Q(3,4) Q(x,9) Lecture 1.2 - Equivalence, and Predicate Logic

x B(x)? The Universal Quantifier Another way of changing a predicate into a proposition. Suppose P(x) is a predicate on some universe of discourse. The universal quantifier of P(x) is the proposition: “P(x) is true for all x in the universe of discourse.” We write it x P(x), and say “for all x, P(x)” x P(x) is TRUE if P(x) is true for every single x. x P(x) is FALSE if there is an x for which P(x) is false. Ex: B(x) = “x is carrying a backpack,” x is a set of cs250 students. Lecture 1.2 - Equivalence, and Predicate Logic

A: only a is true B: only b is true C: both are true D: neither is true Universe of discourse is people in this room. The Universal Quantifier - example B(x) = “x is allowed to drive a car” L(x) = “x is at least 16 years old.” Are either of these propositions true? • x (L(x)  B(x)) • x B(x) Lecture 1.2 - Equivalence, and Predicate Logic

x C(x)? The Existential Quantifier Another way of changing a predicate into a proposition. Suppose P(x) is a predicate on some universe of discourse. The existential quantifier of P(x) is the proposition: “P(x) is true for some x in the universe of discourse.” We write it x P(x), and say “for some x, P(x)” x P(x) is TRUE if there is an x for which P(x) is true. x P(x) is FALSE if P(x) is false for every single x. Ex. C(x) = “x has black hair,” x is a set of cs 250 students. Lecture 1.2 - Equivalence, and Predicate Logic

A: only a is true B: only b is true C: both are true D: neither is true Universe of discourse is people in this room. The Existential Quantifier - example B(x) = “x is wearing sneakers.” L(x) = “x is at least 21 years old.” Y(x)= “x is less than 24 years old.” Are either of these propositions true? • x B(x) • x (Y(x)  L(x)) Lecture 1.2 - Equivalence, and Predicate Logic

x (L(x)  F(x)) Universe of discourse is all creatures. x (L(x)  C(x)) x (F(x)  C(x)) Predicates – more examples L(x) = “x is a lion.” F(x) = “x is fierce.” C(x) = “x drinks coffee.” All lions are fierce. Some lions don’t drink coffee. Some fierce creatures don’t drink coffee. Lecture 1.2 - Equivalence, and Predicate Logic

x (B(x)  R(x)) Universe of discourse is all creatures. x (L(x)  H(x)) x (H(x)  R(x)) Predicates – some more example B(x) = “x is a hummingbird.” L(x) = “x is a large bird.” H(x) = “x lives on honey.” R(x) = “x is richly colored.” All hummingbirds are richly colored. No large birds live on honey. Birds that do not live on honey are dully colored. Lecture 1.2 - Equivalence, and Predicate Logic

x (L(x)  H(x)) x (L(x)  H(x)) Quantifier Negation Not all large birds live on honey. x P(x) means “P(x) is true for every x.” What about x P(x) ? Not [“P(x) is true for every x.”] “There is an x for which P(x) is not true.” x P(x) So, x P(x) is the same as x P(x). Lecture 1.2 - Equivalence, and Predicate Logic

x (L(x)  H(x)) x (L(x)  H(x)) Quantifier Negation No large birds live on honey. x P(x) means “P(x) is true for some x.” What about x P(x) ? Not [“P(x) is true for some x.”] “P(x) is not true for all x.” x P(x) So, x P(x) is the same as x P(x). Lecture 1.2 - Equivalence, and Predicate Logic

Quantifier Negation So, x P(x) is the same as x P(x). So, x P(x) is the same as x P(x). General rule: to negate a quantifier, move negation to the right, changing quantifiers as you go. Lecture 1.2 - Equivalence, and Predicate Logic

Some quick questions • p T =? (what law?) • p  T =? (what law?) • (p  q) = ? (what law?) • P(x) = “x >3” • P(x) is a proposition: yes or no? • P(3) is a proposition: yes or no? If yes, what is its value? • P(4) is True or False? • If the universe of discourse is all natural numbers, what is the value of • x P(x) • x P(x) Lecture 1.2 - Equivalence, and Predicate Logic

Today’s Reading and Next Lecture • Rosen 1.3 and 1.4 • Please start solving the exercises at the end of each chapter section. They are fun. • Please read 1.4 and 1.5 in preparation for the next lecture Lecture 1.2 - Equivalence, and Predicate Logic

Lecture 1.2: Equivalences, and Predicate Logic*