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."
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."
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."
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."
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."
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."
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."
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."
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."
Statement:
"Everyone loves themselves."
Representation:
∀x (L(x, x))
Translation:
"For every person x, x loves themself."
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."