Discrete Structures Chapter 3: The Logic of Quantified Statements 3.1 Predicates and Quantified Statements I … it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics. – A. N. Whitehead, 1861 – 1947 3.1 Predicates and Quantified Statements I

3.1 Predicates and Quantified Statements I Definitions Predicate A predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables. Domain The domain of a predicate variable is the set of all values that may be substituted in place of the variable. Premises All statements in an argument and all statement forms in an argument form are called premises except for the last one.. 3.1 Predicates and Quantified Statements I

3.1 Predicates and Quantified Statements I Example – pg. 106 # 3 Let P(x) be the predicate “x > 1/x”. Write P(2), P(1/2), P(-1), P(-1/2), and P(-8), and indicate which of these statements are true and which are false. Find the truth set of P(x) if the domain of x is , the set of all real numbers. If the domain is the set + of all positive real numbers, what is the truth set of P(x)? 3.1 Predicates and Quantified Statements I

3.1 Predicates and Quantified Statements I Quantifiers We can obtain statements from predicates by adding quantifiers or words that refer to quantities. The symbol  denotes “for all” and is called the universal quantifier. The symbol  denotes “there exists” and is called the existential quantifier. 3.1 Predicates and Quantified Statements I

3.1 Predicates and Quantified Statements I Definitions Let Q(x) be a predicate and D the domain of x: Universal Statement A universal statement is a statement of the form “xD, Q(x).” It is defined to be true iff Q(x) is true for all x in D. It is defined to be false iff Q(x) is false for at least one x in D. A value for x for which Q(x) is false is called a counterexample to the universal statement. Existential Statement A existential statement is a statement of the form “xD such that Q(x).” It is defined to be true iff Q(x) is true for at least one x in D. It is false iff Q(x) is false for all x in D. 3.1 Predicates and Quantified Statements I

Formal Versus Informal When working with mathematics, it is important to be able to translate between formal (symbols) to informal (words) and visa versa. 3.1 Predicates and Quantified Statements I

3.1 Predicates and Quantified Statements I Example – pg. 107 # 15 Rewrite the following statements informally in at least two different ways without using variables or quantifiers.  rectangles x, x is a quadrilateral.  a set A such that A has 16 subsets. 3.1 Predicates and Quantified Statements I

3.1 Predicates and Quantified Statements I Example – pg. 108 #28 Rewrite each statement without using quantifiers or variables. Indicate which are true and which are false, and justify your answers as best as you can. Let the domain of x be the set D of objects discussed in mathematics courses, let Real(x) be “x is a real number”, let Pos(x) be “x is a positive real number”, let Neg(x) be “x is a negative real number”, and let Int(x) be “x is an integer.” Pos(0) x, Real(x)  Neg(x)  Pos(-x). x, Int(x)  Real(x).  x s.t. Real (x)  ~Int (x). 3.1 Predicates and Quantified Statements I