Predicates and Quantifiers Dr. Yasir Ali

How do you translate the statement: All birds can fly How do you translate the statement: All birds can fly. If D= Set of all creatures Take 𝐵(𝑥)= 𝑥 is a Bird. 𝐹(𝑥)= 𝑥 can fly. ∀ 𝑥, 𝐵(𝑥)→𝐹(𝑥) We can not say, ∀ 𝑥, 𝐵(𝑥)∧𝐹(𝑥) Since it means that All creatures are bird and they can fly, which is not a correct translation of original statement.

How do you translate the statement: Some birds can fly How do you translate the statement: Some birds can fly. If D= Set of all creatures Take 𝐵(𝑥)= 𝑥 is a Bird. 𝐹(𝑥)= 𝑥 can fly. ∃ 𝑥, 𝐵(𝑥)∧𝐹(𝑥) We can not say, ∃ 𝑥, 𝐵(𝑥)→𝐹(𝑥) Since this statement will be true if 𝑥 is a frog. That is this statement will be true if 𝑥 is not a bird because in that case F→𝐹 and 𝐹→𝑇 both are the true statements.

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Everyone in your class has a cellular phone. b) Somebody in your class has seen a foreign movie. c) There is a person in your class who cannot swim. d) All students in your class can solve quadratic equations. e) Some student in your class does not want to be rich.

Precedence of Quantifiers The quantifiers ∀ and ∃ have higher precedence than all logical operators from propositional calculus. For example, ∀𝑥𝑃(𝑥) ∨ 𝑄(𝑥) is the disjunction of ∀𝑥𝑃(𝑥) and 𝑄(𝑥). In other words, it means (∀𝑥𝑃(𝑥)) ∨ 𝑄(𝑥) rather than ∀𝑥(𝑃(𝑥) ∨ 𝑄(𝑥)).

Express each of these statements using quantifiers Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the case that.”) a) All dogs have fleas. b) There is a horse that can add. c) Every koala can climb. d) No monkey can speak French. e) There exists a pig that can swim and catch fish.

Translate into formal language. “All lions are fierce Translate into formal language. “All lions are fierce.” “Some lions do not drink coffee.” “Some fierce creatures do not drink coffee.”

Multiple Quantified Statements Assume that the domain for the variables 𝑥 and 𝑦 consists of all real numbers. The statement ∀𝑥 ∀𝑦(𝑥 + 𝑦 = 𝑦 + 𝑥) says that 𝑥 + 𝑦 = 𝑦 + 𝑥 for all real numbers 𝑥 and 𝑦. This is the commutative law for addition of real numbers. Translate into English the statement ∀𝑥 ∀𝑦((𝑥 > 0) ∧ (𝑦 < 0) → (𝑥𝑦 < 0)), where the domain for both variables consists of all real numbers. The product of a positive real number and a negative real number is always a negative real number.”

Translate these statements into English, where the domain for each variable consists of all real numbers. a) ∀𝑥∃𝑦(𝑥 < 𝑦) b) ∀𝑥∀𝑦(((𝑥 ≥ 0) ∧ (𝑦 ≥ 0)) → (𝑥𝑦 ≥ 0)) c) ∀𝑥∀𝑦∃𝑧(𝑥𝑦 = 𝑧) Suppose the domain of the propositional function is {1,2,3}. Write out these propositions using disjunctions and conjunctions and determine the truth value of each of these statements. ∀𝑥∃𝑦(𝑥 + 𝑦 = 1) ∀𝑥∀𝑦∃𝑧(𝑧 = (𝑥 + 𝑦))

The Order of Quantifiers Consider the statements: ∀ 𝑝𝑒𝑜𝑝𝑙𝑒 𝑥 ∃ 𝑎 𝑝𝑒𝑟𝑠𝑜𝑛 𝑦 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥 𝑙𝑜𝑣𝑒𝑠 𝑦. ∃ 𝑎 𝑝𝑒𝑟𝑠𝑜𝑛 𝑦 ∀ 𝑝𝑒𝑜𝑝𝑙𝑒 𝑥 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥 𝑙𝑜𝑣𝑒𝑠 𝑦.

Let 𝑄(𝑥, 𝑦) denote “𝑥 + 𝑦 = 0.” What are the truth values of the quantifications ∃𝑦∀𝑥𝑄(𝑥, 𝑦) and ∀𝑥∃𝑦𝑄(𝑥, 𝑦), where the domain for all variables consists of all real numbers? ∃𝑦∀𝑥𝑄(𝑥, 𝑦) denotes the proposition “There is a real number 𝑦 such that for every real number 𝑥, 𝑄(𝑥, 𝑦).” The quantification ∀𝑥∃𝑦𝑄(𝑥, 𝑦) “For every real number 𝑥 there is a real number 𝑦 such that 𝑄(𝑥, 𝑦).”

Quantifications of Two Variables Statement When True? When False? ∀𝑥∀𝑦𝑃(𝑥, 𝑦) ∀𝑦∀𝑥𝑃(𝑥, 𝑦) 𝑃(𝑥, 𝑦) is true for every pair 𝑥, 𝑦. There is a pair 𝑥, 𝑦 for which 𝑃(𝑥, 𝑦) is false. ∀𝑥∃𝑦𝑃(𝑥, 𝑦) For every 𝑥 there is a 𝑦 for which 𝑃(𝑥, 𝑦) is true. There is an x such that 𝑃(𝑥, 𝑦) is false for every 𝑦. ∃𝑥∀𝑦𝑃(𝑥, 𝑦) There is an 𝑥 for which 𝑃(𝑥, 𝑦) is true for every 𝑦. For every 𝑥 there is a 𝑦 for which 𝑃(𝑥, 𝑦) is false. ∃𝑥∃𝑦𝑃(𝑥, 𝑦) ∃𝑦∃𝑥𝑃(𝑥, 𝑦) There is a pair 𝑥, 𝑦 for which 𝑃(𝑥, 𝑦) is true. 𝑃(𝑥, 𝑦) is false for every pair 𝑥, 𝑦.

Express each of these mathematical statements using predicates, quantifiers, logical connectives, and mathematical operators. a) The product of two negative real numbers is positive. b) The difference of a real number and itself is zero. d) A negative real number does not have a square root that is a real number.

Let F(x, y) be the statement “x can fool y,” where the domain consists of all people in the world. Use quantifiers to express each of these statements. a) Everybody can fool Fred. b) Evelyn can fool everybody. c) Everybody can fool somebody. d) There is no one who can fool everybody. e) Everyone can be fooled by somebody. f ) No one can fool both Fred and Jerry. g) Nancy can fool exactly two people.

Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers. a) ∃𝑥∀𝑦(𝑥 + 𝑦 = 𝑦) b) ∀𝑥∀𝑦(((𝑥 ≥ 0) ∧ (𝑦 < 0)) → (𝑥 −𝑦 > 0)) c) ∃𝑥∃𝑦(((𝑥 ≤ 0) ∧ (𝑦 ≤ 0)) ∧ (𝑥 −𝑦 > 0)) d) ∀𝑥∀𝑦((𝑥 = 0) ∧ (𝑦 = 0) ↔ (𝑥𝑦 = 0))

Rules of inference for quantified statements Rule of Inference Name ∀xP(x) ∴ P(c) Universal instantiation P(c) for an arbitrary c ∴ ∀xP(x) Universal generalization ∃xP(x) ∴ P(c) for some element c Existential instantiation P(c) for some element c ∴ ∃xP(x) Existential generalization

Show that the following argument is valid or not: “Ali, a student in this class, knows how to write programs in JAVA. Everyone who knows how to write programs in JAVA can get a high-paying job. Therefore, someone in this class can get a high-paying job.” Everyone enrolled in the university has lived in a dormitory. Mia has never lived in a dormitory. Therefore, Mia is not enrolled in the university.