Answer
a) ∀x(Student(x) -> Likes(x,math) = 1)
Negation: ∃x(Student(x) & Likes(x,math) = 0)
English: There exists at least one student in this class who does not like mathematics.
b) ∃x(Student(x) & ∀y(Computer(y) -> Seen(x,y) = 0))
Negation: ∀x(Student(x) -> ∃y(Computer(y) & Seen(x,y) = 1))
English: For every student in this class, there exists at least one computer that the student has seen.
c) ∃x(Student(x) & ∀y(MathCourse(y) -> Taken(x,y) = 1))
Negation: ∀x(Student(x) -> ∃y(MathCourse(y) & Taken(x,y) = 0))
English: For every student in this class, there exists at least one mathematics course offered at this school that the student has not taken.
d) ∃x(Student(x) & ∀y(Building(y) -> ∃z(Room(z) & In(z,y) & Been(x,z) = 1)))
Negation: ∀x(Student(x) -> ∃y(Building(y) & ∀z(Room(z) -> In(z,y) -> Been(x,z) = 0)))
English: For every student in this class, there exists at least one building on campus such that the student has never been in any room of that building.
Work Step by Step
a) The original statement is "Every student in this class likes mathematics." 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 "Likes(x,math)" which states that x likes mathematics. The statement says that for every student x in this class, x likes mathematics (∀x(Student(x) -> Likes(x,math) = 1)).
The negation of this statement is "There exists at least one student in this class who does not like mathematics." 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 "Likes(x,math) = 0" which states that x does not like mathematics. The negation says that there exists at least one student x in this class, such that x does not like mathematics (∃x(Student(x) & Likes(x,math) = 0)).
b) The original statement is "There is a student in this class who has never seen a computer." This statement can be expressed using the quantifier "there exists" (∃) and the predicates "Student(x)" which states that x is a student in this class, "Computer(y)" which states that y is a computer, and "Seen(x,y) = 0" which states that x has never seen y. The statement says that there exists at least one student x in this class who has never seen a computer (∃x(Student(x) & ∀y(Computer(y) -> Seen(x,y) = 0)).
The negation of this statement is "For every student in this class, there exists at least one computer that the student has seen." This negation can be expressed by using the quantifier "for all" (∀) and the predicates "Student(x)" which states that x is a student in this class, "Computer(y)" which states that y is a computer, and "Seen(x,y) = 1" which states that x has seen y. The negation says that for every student x in this class, there exists at least one computer y such that x has seen y (∀x(Student(x) -> ∃y(Computer(y) & Seen(x,y) = 1)).
c) The original statement is "There is a student in this class who has taken every mathematics course offered at this school." This statement can be expressed using the quantifier "there exists" (∃) and the predicates "Student(x)" which states that x is a student in this class, "MathCourse(y)" which states that y is a mathematics course offered at this school, and "Taken(x,y) = 1" which states that x has taken y. The statement says that there exists at least one student x in this class who has taken every mathematics course offered at this school (∃x(Student(x) & ∀y(MathCourse(y) -> Taken(x,y) = 1)).
The negation of this statement is "For every student in this class, there exists at least one mathematics course offered at this school that the student has not taken." This negation can be expressed by using the quantifier "for all" (∀) and the predicates "Student(x)" which states that x is a student in this class, "MathCourse(y)" which states that y is a mathematics course offered at this school, and "Taken(x,y) = 0" which states that x has not taken y. The negation says that for every student x in this class, there exists at least one mathematics course y offered at this school such that x has not taken y (∀x(Student(x) -> ∃y(MathCourse(y) & Taken(x,y) = 0)).
d) The original statement is "There is a student in this class who has been in at least one room of every building on campus." This statement can be expressed using the quantifier "there exists" (∃) and the predicates "Student(x)" which states that x is a student in this class, "Building(y)" which states that y is a building on campus, "Room(z)" which states that z is a room, "In(z,y)" which states that z is in building y and "Been(x,z) = 1" which states that x has been in room z. The statement says that there exists at least one student x in this class such that for every building y on campus there is at least one room z that is in y and x has been in z (∃x(Student(x) & ∀y(Building(y) -> ∃z(Room(z) & In(z,y) & Been(x,z) = 1)))
The negation of this statement is "For every student in this class, there exists at least one building on campus such that the student has never been in any room of that building." This negation can be expressed by using the quantifier "for all" (∀) and the predicates "Student(x)" which states that x is a student in this class, "Building(y)" which states that y is a building on campus, "Room(z)" which states that z is a room, "In(z,y)" which states that z is in building y and "Been(x,z) = 0" which states that x has never been in room z. The negation says that for every student x in this class, there exists at least one building y on campus such that for every room z that is in y, x has never been in z (∀x(Student(x) -> ∃y(Building(y) & ∀z(Room(z) -> In(z,y) -> Been(x,z) = 0))).
