Nested Quantifiers

1
Nested Quantifiers

• CS/APMA 202, Spring 2005
• Rosen, section 1.4
• Aaron Bloomfield

2
Multiple quantifiers

• You can have multiple quantifiers on a statement
• ?x?y P(x, y)
• For all x, there exists a y such that P(x,y)
• Example ?x?y (xy 0)
• ?x?y P(x,y)
• There exists an x such that for all y P(x,y) is
  true
• ?x?y (xy 0)

3
Order of quantifiers

• ?x?y and ?x?y are not equivalent!
• ?x?y P(x,y)
• P(x,y) (xy 0) is false
• ?x?y P(x,y)
• P(x,y) (xy 0) is true

4
Negating multiple quantifiers

• Recall negation rules for single quantifiers
• ?x P(x) ?x P(x)
• ?x P(x) ?x P(x)
• Essentially, you change the quantifier(s), and
  negate what its quantifying
• Examples
• (?x?y P(x,y))
• ?x ?y P(x,y)
• ?x?y P(x,y)
• (?x?y?z P(x,y,z))
• ?x?y?z P(x,y,z)
• ?x?y?z P(x,y,z)
• ?x?y?z P(x,y,z)

5
Negating multiple quantifiers 2

• Consider (?x?y P(x,y)) ?x?y P(x,y)
• The left side is saying for all x, there exists
  a y such that P is true
• To disprove it (negate it), you need to show that
  there exists an x such that for all y, P is
  false
• Consider (?x?y P(x,y)) ?x?y P(x,y)
• The left side is saying there exists an x such
  that for all y, P is true
• To disprove it (negate it), you need to show that
  for all x, there exists a y such that P is false

6
Translating between English and quantifiers

• Rosen, section 1.4, question 20
• The product of two negative integers is positive
• ?x?y ((xlt0) ? (ylt0) ? (xy gt 0))
• Why conditional instead of and?
• The average of two positive integers is positive
• ?x?y ((xgt0) ? (ygt0) ? ((xy)/2 gt 0))
• The difference of two negative integers is not
  necessarily negative
• ?x?y ((xlt0) ? (ylt0) ? (x-y0))
• Why and instead of conditional?
• The absolute value of the sum of two integers
  does not exceed the sum of the absolute values of
  these integers
• ?x?y (xy x y)

7
Translating between English and quantifiers

• Rosen, section 1.4, question 24
• ?x?y (xy y)
• There exists an additive identity for all real
  numbers
• ?x?y (((x0) ? (ylt0)) ? (x-y gt 0))
• A non-negative number minus a negative number is
  greater than zero
• ?x?y (((x0) ? (y0)) ? (x-y gt 0))
• The difference between two non-positive numbers
  is not necessarily non-positive (i.e. can be
  positive)
• ?x?y (((x?0) ? (y?0)) ? (xy ? 0))
• The product of two non-zero numbers is non-zero
  if and only if both factors are non-zero

8
End of lecture on 8 February 2005
9
Rosen, section 1.4 question 30

• Rewrite these statements so that the negations
  only appear within the predicates
• ??y?x P(x,y)
• ?y??x P(x,y)
• ?y?x ?P(x,y)
• ??x?y P(x,y)
• ?x??y P(x,y)
• ?x?y ?P(x,y)
• ??y (Q(y) ? ?x ?R(x,y))
• ?y ?(Q(y) ? ?x ?R(x,y))
• ?y (?Q(y) ? ?(?x ?R(x,y)))
• ?y (?Q(y) ? ?x R(x,y))

10
Rosen, section 1.4 question 31

• Express the negations of each of these statements
  so that all negation symbols immediately precede
  predicates.
• ?x?y?z T(x,y,z)
• ?(?x?y?z T(x,y,z))
• ??x?y?z T(x,y,z)
• ?x??y?z T(x,y,z)
• ?x?y??z T(x,y,z)
• ?x?y?z ?T(x,y,z)
• ?x?y P(x,y) ? ?x?y Q(x,y)
• ?(?x?y P(x,y) ? ?x?y Q(x,y))
• ??x?y P(x,y) ? ??x?y Q(x,y)
• ?x??y P(x,y) ? ?x??y Q(x,y)
• ?x?y ?P(x,y) ? ?x?y ?Q(x,y)

11
Quick survey

• I felt I understood the material in this slide
  set
• Very well
• With some review, Ill be good
• Not really
• Not at all

12
Quick survey

• The pace of the lecture for this slide set was
• Fast
• About right
• A little slow
• Too slow

13
Quick survey

• How interesting was the material in this slide
  set? Be honest!
• Wow! That was SOOOOOO cool!
• Somewhat interesting
• Rather borting
• Zzzzzzzzzzz