Lesson 3: Propositional Functions, Quantifiers

1
Lesson 3 Propositional Functions, Quantifiers

• Objectives
• State the truth value of a propositional function
  given its arguments
• Determine truth values of universal
  quantifications and existential quantifications
• Explain the difference between a propositional
  function and a proposition
• Prove existential quantification
• Disprove universal quantification
• Translate conditionals in English to mathematical
  notation
• Translate mathematical notation into English
  statements
• Negate nested quantifiers

• Outline
• Propositional Functions
• Quantifiers
• Nested Quantifiers
• Reading Section 1.3, 1.4
• Problems Due 1/25
• 1.3 10, 12, 13, 21bcd, 34, 48
• 1.4 2, 4ace, 7bc, 12c-f, 16bc

2
Negations

• Every student at VU has a GPA gt 2.0
• There is a student who does not have a GPA gt
  2.0
• There is a person who can lift 500 Kg
• Everyone in the world is unable to lift 500 Kg

3
Proofs of quantifiers

• P(x) Person x is of Irish descent
• (universe of discourse is people in this room)
• ?xP(x)
• ?xP(x)
• To prove universal false, show counterexample
• To prove existential true, show example

4
Translations

• S(x) x is a sophomore x ? VU students
• E(x) x majors in ECE
• T(x,y) x is taking course y y ? Courses at VU
• State in English
• S(Alan)
• ?xE(x)
• ?xT(x, CORE110)
• ?xT(x,y)

5
Translations

• ?yT(Ed, y)
• ?y ? T(Sue,y)
• No student takes BSKT100.
• There is a course that all ECE sophomores are
  taking

6
Example

• P(x) x is a prime number
• E(x) x is even universe integers
• ?xP(x) ? E(x)
• All prime numbers are odd
• No even integer is odd

7
Nested Quantifiers

• Quantifiers can occur within the scope of other
  quantifiers
• T(x,y) student x has taken class y
• ?y?xT(x,y)
• ?x?yT(x,y)

8
Example

• For all integers x,y,z
• x y z z x y
• For every number x there is a number y such that
  xy1

9
Translations

• T(x) x is a student x ? all people
• S(x,y) x shops in store y y ? all stores
• ?yS(Zsa Zsa, y)
• ?y?xT(x) ? ?S(x,y)
• ?y?xT(x)?S(x,y)
• ?x1?y?x2S(x1,y)?(x1?x2) ? ?S(x2,y)

10
Translation

• Ed shops at Sears
• No stores have no students who will shop there
• Some stores have only students as shoppers

11
Negation of Nested Quantifiers

• There is no largest number.
• Given a number x, there exists a number y which
  is greater than x
• ?x?y(ygtx)
• It is not the case that there exists a number x
  such that x is greater or equal to all other
  numbers y
• ??x?y(xy)

12
Order of Quantifiers

• Q(x,y) x y 0
• ?x?y Q(x,y)
• ?y?x Q(x,y)
• (review Table 1, pg. 50)

13
Brain Food

• No soldier shall, in time of peace, be quartered
  in any house, without the consent of the owner,
  nor in time of war, but in a manner to be
  prescribed by law.
• Create two implications directly from this
  statement by breaking it up into two conditions
  peacetime and wartime. State them in English.
• Under what conditions can a soldier be quartered
  in my house?

14
Extras
15
Extras