Statements containing multiple quantifiers

1
Statements containing multiple quantifiers

• Ex 1) ?x?R, ?y?Z such that x-ylt1.
• 2) For any building x in the city
• there is a fire-station y such that the
  distance between x and y
• is at most 2 miles.
• 3) ?x?Z such that ?y?3,5, xlty.
• 4) There is a student who solved all
  the problems of the exam correctly.

2
Truth values of multiply quantified statements

• Ex ? Students Joe, Ann, Bob, Dave
• ? 2 groups of languages
• Asian languagesJapanese,Chinese,Korean
• European languagesFrench, German,
  Italian, Spanish.
• ? Joe speaks Italian and French
• Ann speaks German, French and Japanese
• Bob speaks Spanish, Italian and Chinese
• Dave speaks Japanese and Korean.

3
Truth values of multiply quantified statements

• Ex(cont.) Determine truth values of the
  following statements
• 1) ? a student S s.t. ? language L,
• S speaks L.
• 2) ? a student S s.t. for ? language group G ?
  L in G s.t. S speaks L.
• 3) ? a language group G s.t. for ? student S ?
  L in G s.t. S speaks L.

4
Negating multiply quantified statements

• Example
• The negation of for ?x?R, ?y?R s.t. y2x
• is logically equivalent to
• ?x?R s.t. for ?y?R, y2?x.
• Generally,
• the negation of ?x, ?y s.t. P(x,y)
• is logically equivalent to
• ?x s.t. ?y, P(x,y)

5
Negating multiply quantified statements

• Example
• The negation of ? x?R s.t. ?y?Z, xgty
• is logically equivalent to
• ?x?R ?y?Z s.t. xy.
• Generally,
• the negation of ?x s.t. ?y, P(x,y)
• is logically equivalent to
• ?x ?y s.t. P(x,y)

6
The Relation among ?, ?, ?, ?

• Let Q(x) be a predicate
• Dx_1, x_2, , x_n be the domain of x.
• Then
• ? ?x?D, Q(x) is logically equivalent to
• Q(x_1) ? Q(x_2) ? ? Q(x_n)
• ? ?x?D, Q(x) is logically equivalent to
• Q(x_1) ? Q(x_2) ? ? Q(x_n) .