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Discrete Structures

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.

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Discrete Structures

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  1. 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

  2. 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. 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

  4. 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

  5. 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 iffQ(x) is true for all x in D. It is defined to be false iffQ(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 iffQ(x) is true for at least one x in D. It is false iffQ(x) is false for all x in D. 3.1 Predicates and Quantified Statements I

  6. 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

  7. 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

  8. 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

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