Nested Quantifiers in Discrete Structures: Understanding Multiple Quantifiers, Lecture notes of Computer Science

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CSC102 - Discrete Structures By Mahwish Waqas

Department Of Computer Science, CUI

Lahore Campus

Course Outline

• Nested Quantifiers

Nested Quantifiers

• “For all 𝑥, there exists a 𝑦 such that 𝑃(𝑥, 𝑦)”.
• Example:
• ∀𝑥∃𝑦 (𝑥 + 𝑦 = 0 ) where 𝑥 and 𝑦 are integers
• 𝑥𝑦 𝑃(𝑥, 𝑦)
• There exists an x such that for all 𝑦, 𝑃(𝑥, 𝑦) is true”
• Example: 𝑥𝑦 𝑥 × 𝑦 = 0
• ∀𝑥∀𝑦∀𝑧 𝑥 + 𝑦 + 𝑧 = 𝑥 + 𝑦 + 𝑧
• THINK QUANTIFICATION AS LOOPS

Nested Quantifiers Example

• Let Domain of 𝑥 is the students in this class Doamin of 𝑦 is the courses in software engineering 𝑄 𝑥, 𝑦 = “𝑥 takes course 𝑦”, true when 𝑥 takes course 𝑦, otherwise false. Translate the following logical expression:
• ∀𝑥∀𝑦𝑄 𝑥, 𝑦
• ∃𝑥∃𝑦𝑄(𝑥, 𝑦)
• ∀𝑥∃𝑦𝑄 𝑥, 𝑦
• ∃𝑥∀𝑦𝑄(𝑥, 𝑦)

Predicates - the meaning of multiple quantifiers

Statement When True? When False? ∀ x ∀ yP(x, y) ∀ y ∀ xP(x, y) P(x, y) is true for every pair x, y. There is a pair x, y for which P(x, y) is false. ∀ x ∃ yP(x, y) For every x there is a y for which P(x, y) is true. There is an x such that P(x, y) is false for every y. ∃ x ∀ yP(x, y) There is an x for which P(x, y) is true for every y. For every x there is a y for which P(x, y) is false. ∃ x ∃ yP(x, y) ∃ y ∃ xP(x, y) There is a pair x, y for which P(x, y) is true. P(x, y) is false for every pair x, y.

Example

• Let Q(x, y): “ x + y = 0 ” What are the truth values of the quantifications ∃ y ∀ xQ(x, y) and ∀ x ∃ yQ(x,y) , where the domain for all variables consists of all real numbers?

Example

• The quantification ∀ x ∃ yQ(x, y) denotes the proposition “For every real number x there is a real number y such that Q(x, y) .”
• Given a real number x , there is a real number y such that x + y = 0; namely, y = − x.
• Hence, the statement ∀ x ∃ yQ(x, y) is true.

Order of Quantifiers

• xy and xy are not equivalent!
• xy P(x,y)
  • P(x,y) = (x+y == 0) is false
• xy P(x,y)
  • P(x,y) = (x+y == 0) is true

Translating between English and Quantifiers

• Translate the statement “The sum of two positive integers is always positive” into a logical expression.

Translating between English and Quantifiers

• Translate the statement “The sum of two positive integers is always positive” into a logical expression.
• Solution:
• First rewrite it so that the implied quantifiers and a domain are shown: “For every two integers, if these integers are both positive, then the sum of these integers is positive.”
• Next, introduce the variables x and y to obtain “For all positive integers x and y , x + y is positive.”
• Statement is ∀ x ∀ y((x > 0 ) ∧ (y > 0 ) → (x + y > 0 )),
• where the domain for both variables consists positive integers.

Translating between English and Quantifiers

• Translate the following statement into English ∀ x(C(x) ∨ ∃ y(C(y) ∧ F(x, y))) where C(x): “ x has a computer,” F(x, y): “ x and y are friends,” The domain for both x and y consists of all students in your school.
• Solution:
• The statement says that for every student x in your school, x has a computer or there is a student y such that y has a computer and x and y are friends.
• In other words, every student in your school has a computer or has a friend who has a computer.

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 it’s 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)

Chapter Reading

• Chapter 1 , Kenneth H. Rosen, Discrete Mathematics and Its Applications, Section 1.

Chapter Exercise (For Practice)

• Question # 1, 2, 3, 4, 8, 23, 24, 25, 26, 27, 30, 31, 39, 41