Recording-2025-02-09T21:06:21.236Z

Translating Nested Quantifiers into English

Example 1: Students at TCNJ

• Let:

  • c(x): x has a computer

  • f(x, y): x and y are friends

  • Domain: All students at TCNJ

• Statement:

  • ∀x, ∃y (c(x) ∨ (c(y) ∧ f(x, y)))

• Translation:

  • "Every student at TCNJ has a computer or has a friend who has a computer."

Example 2: Friends of a Student

• Let:

  • f(x, y): x and y are friends

• Statement:

  • ∃x, ∀y, ∀z (f(x, y) ∧ f(x, z) ∧ (y ≠ z)) → ¬f(y, z)

• Translation:

  • "There exists a student such that any two of their friends are not friends with each other."

Example 3: Positive Integers

• Statement:

  • "The sum of two positive integers is also positive."

• Representation:

  • ∀x, ∀y (x > 0 ∧ y > 0) → (x + y > 0)

• Translation:

  • "For all positive integers x and y, if x and y are positive, then x plus y is also positive."

Example 4: Woman and Flights

• Statement:

  • "There is a woman who has taken a flight on every airline."

• Representation:

  • ∃w (∀a (∃f (p(w, f) ∧ q(f, a))))

  • where:

    • p(w, f): w has taken flight f

    • q(f, a): flight f is on airline a

• Translation:

  • "There exists a woman such that for every airline, there is a flight that she has taken on that airline."

Example 5: Brother and Sibling Relationship

• Statement:

  • "Brothers are siblings."

• Representation:

  • ∀x, ∀y (b(x, y) → s(x, y))

• Explanation:

  • "For all x and y, if x is the brother of y, then x is a sibling of y."

Example 6: Symmetric Property of Siblinghood

• Statement:

  • "Siblinghood is symmetric."

• Representation:

  • ∀x, ∀y (s(x, y) → s(y, x))

• Translation:

  • "If x is y's sibling, then y is x's sibling."

Example 7: Everyone Loves Somebody

• Statement:

  • "Everybody loves somebody."

• Representation:

  • ∀x, ∃y (L(x, y))

• Translation:

  • "For every person x, there exists a person y such that x loves y."

Example 8: Loved by Everyone

• Statement:

  • "There is someone who is loved by everyone."

• Representation:

  • ∃y (∀x (L(x, y)))

• Translation:

  • "There exists a person y such that for all x, x loves y."

Example 9: Someone Loves Someone

• Statement:

  • "There is someone who loves someone."

• Representation:

  • ∃x, ∃y (L(x, y))

• Translation:

  • "There exists a person x and a person y such that x loves y."

Example 10: Everyone Loves Themselves

• Statement:

  • "Everyone loves themselves."

• Representation:

  • ∀x (L(x, x))

• Translation:

  • "For every person x, x loves themself."

Negating Nested Quantifiers

• Original Statement:

  • "There is a woman who has taken a flight on every airline."

• Negation:

  • "There does not exist a woman who has taken a flight on every airline."

• Applying De Morgan's Rules:

  • Flip quantifiers and distribute negation:

  • ∀w (∃a (∀f (¬p(w, f) ∨ ¬q(f, a))))

• Translation of Negated Statement:

  • "For every woman, there exists an airline such that for all flights, either the woman has not taken the flight or the flight is not on the airline."