Answer
a) ∀ x (Student(x) -> Taken(x,math) = 2)
Negation: ∃ x (Student(x) & Taken(x,math) ≠ 2)
English: There exists at least one student in this class who has not taken exactly two mathematics classes at this school.
b) ∃ x (Person(x) & ∀ y (Country(y) -> Visited(x,y) OR (y = Libya) ))
Negation: ∀ x (Person(x) -> ∃ y (Country(y) & Visited(x,y) = 0 & y ≠ Libya))
English: For every person, there exists at least one country in the world that they have not visited, and that country is not Libya.
c) ∀ x (Climber(x) -> ∀ y (Mountain(y) -> Climbed(x,y) = 0))
Negation: ∃ x (Climber(x) & ∃ y (Mountain(y) & Climbed(x,y) = 1))
English: There exists at least one climber and at least one mountain in the Himalayas, such that the climber has climbed that mountain.
d) ∀ x (Actor(x) -> (∃ y (Actor(y) & InMovie(x,y) & InMovie(y,Kevin Bacon)) OR (InMovie(x,Kevin Bacon)) )
Negation: ∃ x (Actor(x) & ∀ y (Actor(y) -> !(InMovie(x,y) & (InMovie(y,Kevin Bacon) OR (InMovie(x,Kevin Bacon)))))
English: There exists at least one movie actor who has not been in a movie with Kevin Bacon, and has not been in a movie with someone who has been in a movie with Kevin Bacon.
Work Step by Step
a) The original statement is "Every student in this class has taken exactly two mathematics classes at this school." This statement can be expressed using the quantifier "for all" (∀) and the predicates "Student(x)" which states that x is a student in this class, and "Taken(x,math)" which states that x has taken mathematics classes at this school. The statement says that for every student x in this class, x has taken exactly two mathematics classes at this school (∀ x (Student(x) -> Taken(x,math) = 2)).
The negation of this statement is "There exists at least one student in this class who has not taken exactly two mathematics classes at this school." This negation can be expressed by using the quantifier "there exists" (∃) and the predicates "Student(x)" which states that x is a student in this class, and "Taken(x,math) ≠ 2" which states that x has not taken exactly two mathematics classes at this school. The negation says that there exists at least one student x in this class, such that x has not taken exactly two mathematics classes at this school (∃ x (Student(x) & Taken(x,math) ≠ 2)).
b) The original statement is "Someone has visited every country in the world except Libya." This statement can be expressed using the quantifier "there exists" (∃) and the predicates "Person(x)" which states that x is a person, "Country(y)" which states that y is a country in the world, "Visited(x,y)" which states that x has visited y and "y = Libya" which states that y is Libya. The statement says that there exists at least one person x who has visited every country in the world except Libya (∃ x (Person(x) & ∀ y (Country(y) -> Visited(x,y) OR (y = Libya) ))).
The negation of this statement is "For every person, there exists at least one country in the world that they have not visited, and that country is not Libya." This negation can be expressed by using the quantifier "for all" (∀) and the predicates "Person(x)" which states that x is a person, "Country(y)" which states that y is a country in the world, "Visited(x,y) = 0" which states that x has not visited y and "y ≠ Libya" which states that y is not Libya. The negation says that for every person x, there exists at least one country y in the world such that x has not visited y, and y is not Libya (∀ x (Person(x) -> ∃ y (Country(y) & Visited(x,y) = 0 & y ≠ Libya))).
c) The original statement is "No one has climbed every mountain in the Himalayas." This statement can be expressed using the quantifier "for all" (∀) and the predicates "Climber(x)" which states that x is a climber and "Climbed(x,y) = 0" which states that x has not climbed mountain y. The statement says that for every climber x, for every mountain y in the Himalayas, x has not climbed y (∀ x (Climber(x) -> ∀ y (Mountain(y) -> Climbed(x,y) = 0))).
The negation of this statement is "There exists at least one climber and at least one mountain in the Himalayas, such that the climber has climbed that mountain." This negation can be expressed by using the quantifier "there exists" (∃) and the predicates "Climber(x)" which states that x is a climber, "Mountain(y)" which states that y is a mountain in the Himalayas, and "Climbed(x,y) = 1" which states that x has climbed mountain y. The negation says that there exists at least one climber x and at least one mountain y in the Himalayas such that x has climbed y (∃ x (Climber(x) & ∃ y (Mountain(y) & Climbed(x,y) = 1))).
d) The original statement is "Every movie actor has either been in a movie with Kevin Bacon or has been in a movie with someone who has been in a movie with Kevin Bacon. " This statement can be expressed using the quantifier "for all" (∀) and the predicates "Actor(x)" which states that x is a movie actor, "InMovie(x,y)" which states that x has been in a movie with y, "Actor(y)" which states that y is a movie actor, and "InMovie(y,Kevin Bacon)" which states that y has been in a movie with Kevin Bacon. The statement says that for every movie actor x, x has either been in a movie with Kevin Bacon or has been in a movie with someone who has been in a movie with Kevin Bacon (∀ x (Actor(x) -> (∃ y (Actor(y) & InMovie(x,y) & InMovie(y,Kevin Bacon)) OR (InMovie(x,Kevin Bacon)) )).
The negation of this statement is "There exists at least one movie actor who has not been in a movie with Kevin Bacon, and has not been in a movie with someone who has been in a movie with Kevin Bacon." This negation can be expressed by using the quantifier "there exists" (∃) and the predicates "Actor(x)" which states that x is a movie actor, "InMovie(x,y)" which states that x has been in a movie with y, "Actor(y)" which states that y is a movie actor, and "!(InMovie(x,y) & (InMovie(y,Kevin Bacon) OR (InMovie(x,Kevin Bacon)))" which states that x has not been in a movie with y and y has not been in a movie with Kevin Bacon or x has not been in a movie with Kevin Bacon. The negation says that there exists at least one movie actor x who has not been in a movie with Kevin Bacon, and has not been in a movie with someone who has been in a movie with Kevin Bacon (∃ x (Actor(x) & ∀ y (Actor(y) -> !(InMovie(x,y) & (InMovie(y,Kevin Bacon) OR (InMovie(x,Kevin Bacon)))))).
