Nested Quantifiers

1
Nested Quantifiers

• Section 1.4

2
Recap Section 1.3

• A predicate is generalization of a proposition.
• It is a proposition that contains variables.
• A predicate becomes a proposition if the
  variable(s) contained is(are)
• Assigned specific value(s)
• Quantified
• Universe of discourse the particular domain
  of the variable in a propositional function

3
Recap Section 1.3

• Universal quantification
• P(x) is true for ALL the values of x in the
  universe of discourse.
• ?x P(x).
• Remember ? ? All.
• for all x, P(x)
• If the elements in the universe of discourse can
  be listed, U x1, x2, , xn
• ?x P(x) ? P(x1) ? P(x2) ? ? P(xn)

4
Recap Section 1.3

• Existential quantification
• P(x) is true FOR SOME x in the universe of
  discourse, i.e. EXIST some x
• ?x P(x)
• Remember, ? ? Exist
• for some x, P(x)
• If the elements in the universe of discourse can
  be listed, U x1, x2, , xn
• ?x P(x) ? P(x1) ? P(x2) ? ? P(xn)

5
Recap Section 1.3

• Universal quantifiers usually take implications
• All CS students are smart students.
• ?x C(x) ? S(x)
• Existential quantifiers usually take conjunctions
• Some CS students are smart students.
• ?x C(x) ? S(x)

6
Recap Section 1.3Summary of quantifiers

• ?x P(x)
• True when P(x) is true for every x
• False when P(x) is false for at least one x.
• ?x P(x)
• True when P(x) is true for at least one x
• False when P(x) is false for every x
• Negation changes a universal to an existential
  and vice versa, and negates the predicate
• ?x P(x) ? ?x P(x)
• ?x P(x) ? ?x P(x)

7
Recap Section 1.3Quick examples

• (13b) Determine truth value. UZ
• ? n (2n 3n)
• (16b) Determine truth value UR
• ? n (x2 -1)
• Exercise 17

8
Nested Quantifiers

• Quantifiers that occur within the scope of other
  quantifiers
• Example
• P(x,y) x y 0, UR
• ?x ?y P(x,y)

9
Quantifications of Two Variables

• For all pair x,y P(x,y).
• ?x?y P(x,y) ?y?x P(x,y)
• For every x there is a y such that P(x,y).
• ?x?y P(x,y)
• There is an x such that P(x,y) for all y.
• ?x?y P(x,y)
• There is a pair x,y such that P(x,y).
• ?x?y P(x,y) ?y?x P(x,y)

10
Translating statements with nested quantifiers

• U all real numbers
• ?x ?y (x y y x)
• ?x ?y (x y 0)
• ?x ?y ( (x gt 0) ? (y lt 0) ? (xy lt 0) )
• U all students in cs2813
• C(x) x has a computer
• F(x,y) x and y are friends
• ?x ( C(x) ? ?y (C(y) ? F(x,y)) )

11
Translating Sentences

• U all people
• If a person is female and is a parent, then this
  person is someones mother.
• U all integers
• The sum of two positive integers is positive.

12
Is the order of quantifiers important?

• If the quantifiers are of the same type, then
  order does not matter
• If the quantifiers are of different types, then
  order is important

13
Example

• UR
• Q(x,y) xy0
• What are the truth values for
• ?y ?x Q(x,y) and ?x ?y Q(x,y)
• ?y ?x Q(x,y) There exist at least one y such
  that for every real number x, Q(x,y) is true,
  i.e. xy0.
• FALSE (not for every, only when y is x).
• But
• ?x ?y Q(x,y) For every real number x, there is a
  real number y such that Q(x,y) is true, i.e xy
  0.
• TRUE (for every x when y is x)