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Chapter 2

Chapter 2. The Logic of Quantified Statements. Section 2.3. Multiple Quantifiers . Multiple Quantifiers. A statement may contain multiple quantifiers, ∀,∃ or ∃, ∀. When multiple quantifiers are encountered treat them as they come, “in order” Example

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Chapter 2

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  1. Chapter 2 The Logic of Quantified Statements

  2. Section 2.3 Multiple Quantifiers

  3. Multiple Quantifiers • A statement may contain multiple quantifiers, ∀,∃ or ∃, ∀. • When multiple quantifiers are encountered treat them as they come, “in order” • Example • ∀x ∈D, ∃y ∈E such that x and y satisfy P(x,y) • Find x from D where “a” y from E satisfies property P(x,y) • ∃x ∈D, ∀y ∈Exand y satisfy P(x,y) • Given x in D, any y from E must satisfy P(x,y)

  4. ∀∃ Statement • Show that the following is true: “For all triangles in x, there is a square y such that x and y have the same color.”

  5. ∃∀ Statement • Show that the following is true: “There is a triangle x such that for all circles y, x is to the right of y.”

  6. Interpreting Multiply-Quantified Statements • Write informal statement and give its truth: • ∃item I such that ∀Students S, S chose I • There is an item that all students selected. (true: Pie) • ∀students S and ∀stations Z, ∃ item I in Z such that S chose I. • Every student chose at least one item from every station. (false: Yuen didn’t get a salad).

  7. Informal to Formal Examples • A reciprocal of a real number a is a real number b such that ab = 1. • Translate the following • Every nonzero real number has a reciprocal. • ∀nonzero real number u, ∃a real number v such that u*v = 1. • There is a real number with no reciprocal. • ∃a real number c such that ∀real numbers d, c*d ≠ 1

  8. Example • Translate the following to formal: • “There is a smallest positive number.” • ∃a positive integer m such that ∀ positive integers n, m ≤ n. • Is this true? Is there a positive integer such that it is equal or less than any other positive integer. • Yes…. 1 works.

  9. Negations of Multiply-Quantified Statements • Multiply-Quantified statements may be negated as is the case with simple quantified statements from 2.2. • ~(∀x in D, Q(x) ) ≡∃x ∈D such that ~Q(x) • ~(∃x ∈D such that Q(x)) ≡ ∀x ∈D, ~Q(x) • ~(∀x in D, ∃y in E such that P(x,y)) • First: ∃x in D such that ~(∃y in E such that P(x,y)) • Final: ∃x in D such that ∀y in E, ~P(x,y) • ~(∃x in D, ∀y in E such that P(x,y)) • First: ∀x in D such that ~(∀y in E such that P(x,y)) • Final: ∀x in D such that ∃y in E, such that ~P(x,y)

  10. Example • Write a negation for the following statements: • For all squares x, there is a circle y such that x and y have the same color. • First: ∃a square x such that ~(∃a circle y such that x and y have the same color) • Final: ∃a square x such that ∀circles y, x and y do not have the same color. • Negation is true. sq e is black and no circle is black.

  11. Example • Write a negation for the following statements: • There is a triangle x such that for all squares y, x is to the right of y. • First: ∀triangles x, ~(∀squares y, x is to the right of y) • Final: ∀triangles x, ∃ a square y such that x is not to the right of y.

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