Predicates and Quantifiers Dr. Yasir Ali

1.Predicates 2.Quantifiers a.Universal Quantifiers b.Existential Quantifiers 3.Negation of Quantifiers 4.Universal Conditional Statement 1.Negation of Universal Conditional 5.Multiple Quantifiers 1.Precedence of quantifiers 2.Order of quantifiers 6.Translation

It is frequently necessary to reason logically about statements of the form Everything has the property p or something has the property p. One of the oldest and most famous pieces of logical reasoning, which was known to the ancient Greeks, is an example: All men are mortal. Socrates is a man. Therefore Socrates is mortal. Predicate logic, also called first order logic, is an extension to propositional logic that adds two quantifiers that allow statements like the examples above to be expressed. Everything in propositional logic is also in predicate logic: all the definitions, inference rules, theorems, algebraic laws, etc., still hold.

“Every computer connected to the university network is functioning properly.” No rules of propositional logic allow us to conclude the truth of the statement “MATH3 is functioning properly,” Where MATH3 is one of the computers connected to the university network.

Predicate

Truth Set of a predicate When an element in the domain of the variable of a one- variable predicate is substituted for the variable, the resulting statement is either true or false. The set of all such elements that make the predicate true is called the truth set of the predicate.

Quantifiers One sure way to change predicates into statements is to assign specific values to all their variables. For example, if x represents the number 35, the sentence “x is divisible by 5” is a true statement since 35 = 5· 7. Another way to obtain statements from predicates is to add quantifiers. Quantifiers are words that refer to quantities such as “some” or “all” and tell for how many elements a given predicate is true.

Quantifiers

Existential Quantifier

Rewrite each of the following statements formally. Use quantifiers and variables. 1.All dinosaurs are extinct. 2.Every real number is positive, negative, or zero. 3.No irrational numbers are integers. 4.Some exercises have answers. 5.Some real numbers are rational.

StatementWhen True?When False?

Negation of Quantified Statement Also known as De Morgan’s Law for Quantifiers NegationEquivalent StatementWhen Is Negation True? When False?

What are the negations of the statements “There is an honest politician” and “All Americans eat cheeseburgers”? Let H(x) denote “x is honest” and C(x) denote “x eats cheeseburgers.”

Universal Conditional Statements

Negation

Translating from English into Logical Expressions