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Introduction to Predicates and Quantified Statements I

Introduction to Predicates and Quantified Statements I. Lecture 9 Section 2.1 Wed, Jan 31, 2007. Predicates. A predicate is a sentence that contains a finite number of variables, and becomes a statement when values are substituted for the variables.

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Introduction to Predicates and Quantified Statements I

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  1. Introduction to Predicates and Quantified Statements I Lecture 9 Section 2.1 Wed, Jan 31, 2007

  2. Predicates • A predicate is a sentence that • contains a finite number of variables, and • becomes a statement when values are substituted for the variables. • If today is x, then Discrete Math meets today. • The predicate becomes true when x = Wednesday.

  3. Domains of Predicate Variables • The domainD of a predicate variable x is the set of all values that x may take on. • Let P(x) be the predicate. • x is a free variable. • The truth set of P(x) is the set of all values of xD for which P(x) is true.

  4. Domains of Predicate Variables • In the previous example, we could define the domain of x to be D = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} • Then the truth set of the predicate is S = {Monday, Tuesday, Wednesday, Friday}

  5. The Universal Quantifier • The symbol  is the universal quantifier. • The statement x  S, P(x) means “for all x in S, P(x),” where S  D. • In this case, x is a bound variable, bound by the quantifier .

  6. The Universal Quantifier • The statement is true if P(x) is true for allx in S. • The statement is false if P(x) is false for at leastonex in S.

  7. Examples • Statement • “7 is a prime number” is true. • Predicate • “x is a prime number” is neither true nor false. • Statements • “x {2, 3, 5, 7}, x is a prime number” is true. • “x {2, 3, 6, 7}, x is a prime number” is false.

  8. Examples of Universal Statements • x  {1, …, 10}, x2 > 0. • x  {1, …, 10}, x2 > 100. • x  R, x3 – x  0. • x  R, y  R, x2 + xy + y2  0. • x  , x2 > 100.

  9. Identities • Algebraic identities are universal statements. • When we write the algebraic identity (x + 1)2 = x2 + 2x + 1 what we mean is x  R, (x + 1)2 = x2 + 2x + 1.

  10. Identities • DeMorgan’s Law ~(p q)  (~p)  (~q) is equivalent to p, q  {T, F}, ~(p q) = (~p)  (~q).

  11. The Existential Quantifier • The symbol  is the existential quantifier. • The statement x  S, P(x) means “there exists x in S such that P(x),” where S  D. • x is a bound variable, bound by the quantifier .

  12. The Existential Quantifier • The statement is true if P(x) is true for at least onex in S. • The statement is false if P(x) is false for allx in S.

  13. Examples of Universal Statements • x  {1, …, 10}, x2 > 0. • x  {1, …, 10}, x2 > 100. • x  R, x3 – x  0. • x  R, y  R, x2 + xy + y2  0. • x  , x2 > 100.

  14. Solving Equations • An algebraic equation has a solution if there exists a value for x that makes it true. x  R, (x + 1)2 = x2 + x + 2.

  15. Boolean Expressions • A boolean expression • Is not a contradiction if there exist truth values for its variables that make it true. • Is not a tautology if there exists truth values for its variables that make if false. • The expression p q  p  q. is neither a contradiction nor a tautology.

  16. Universal Conditional Statements • The statement x  S, P(x)  Q(x) is a universal conditional statement. • Example: x  R, if x > 0, then x + 1/x  2. • Example: n  N,if n  13, then (½)n2 + 3n – 4 > 8n + 12.

  17. Universal Conditional Statements • Suppose that predicates P(x) and Q(x) have the same domain D. • If P(x)  Q(x) for all x in the truth set of P(x), then we write P(x)  Q(x).

  18. Examples • x > 0  x + 1/x  2. • n is a multiple of 6  n is a multiple of 2 and n is a multiple of 3. • n is a multiple of 8  n is a multiple of 2 and n is a multiple of 4.

  19. Universal Biconditional Statements • Again, suppose that predicates P(x) and Q(x) have the same domain D. • If P(x)  Q(x) for all x in the truth set of P(x), then we write P(x)  Q(x).

  20. Examples • Which of the following are true? • x > 0  x + 1/x  2. • n is a multiple of 6  n is a multiple of 2 and n is a multiple of 3. • n is a multiple of 8  n is a multiple of 2 and n is a multiple of 4.

  21. Solving Algebraic Equations • Steps in solving algebraic equations are equivalent to universal conditional statements. • The step x + 3 = 8  x = 5 is justified by the statement x + 3 = 8  x = 5.

  22. Solving Algebraic Equations • An algebraic step from P(x) to Q(x) is reversible only if P(x)  Q(x). • Reversible steps • Adding 1 to both sides. • Taking logarithms. • Not reversible steps • Squaring both sides. • Multiplying by an arbitrary constant.

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