Answer
a) ∀ x (Lost(x) → Lost(x) ≤ 1000)
Negation: ∃ x (Lost(x) & Lost(x) > 1000)
English: There is at least one person who has lost more than one thousand dollars playing the lottery.
b) ∃ x (Student(x) & Chat(x) = 1)
Negation: ∀ x (Student(x) -> Chat(x) ≠ 1)
English: For every student in this class, they have chatted with not exactly one other student.
c) ∀ x (Student(x) -> ∀ y (Student(y) -> (Sent(x,y) → Sent(x,y) ≠ 2)))
Negation: ∃ x (Student(x) & ∃ y (Student(y) & Sent(x,y) = 2) )
English: There exists at least one student in this class who sent e-mail to exactly two other students in this class.
d) ∃ x (Student(x) & ∀ y (Exercise(y) -> Solved(x,y)) )
Negation: ∀ x (Student(x) -> ∃ y (Exercise(y) & !Solved(x,y)))
English: For every student, there exists at least one exercise in this book that the student has not solved.
e) ∀ x (Student(x) -> ∀ y (Section(y) -> !(∃ z (Exercise(z) & Solved(x,z) & In(z,y))))
Negation: ∃ x (Student(x) & ∃ y (Section(y) & ∃ z (Exercise(z) & Solved(x,z) & In(z,y))))
English: There exists at least one student and at least one section in this book, such that the student has solved at least one exercise in that section.
Work Step by Step
a) The original statement is "No one has lost more than one thousand dollars playing the lottery." This statement can be expressed using the quantifier "for all" (∀) and the predicate "Lost(x)" which states that the person x has lost some amount of money playing the lottery. The statement says that for all x, if x has lost money playing the lottery, then the amount lost is less than or equal to 1000 dollars (Lost(x) → Lost(x) ≤ 1000).
The negation of this statement is "There is at least one person who has lost more than one thousand dollars playing the lottery." This negation can be expressed by using the quantifier "there exists" (∃) and the predicate "Lost(x)" which states that the person x has lost some amount of money playing the lottery. The negation says that there exists at least one person x such that x has lost money playing the lottery and the amount lost is greater than 1000 dollars (∃ x (Lost(x) & Lost(x) > 1000)).
b) The original statement is "There is a student in this class who has chatted with exactly one other student." This statement can be expressed using the quantifier "there exists" (∃) and the predicate "Student(x)" which states that x is a student in this class, and the predicate "Chat(x) = 1" which states that x has chatted with exactly one other student. The statement says that there exists at least one student x in this class, such that x has chatted with exactly one other student (∃ x (Student(x) & Chat(x) = 1)).
The negation of this statement is "For every student in this class, they have chatted with not exactly one other student." This negation can be expressed by using the quantifier "for all" (∀) and the predicate "Student(x)" which states that x is a student in this class, and the predicate "Chat(x) ≠ 1" which states that x has not chatted with exactly one other student. The negation says that for every student x in this class, x has not chatted with exactly one other student (∀ x (Student(x) -> Chat(x) ≠ 1)).
c) The original statement is "No student in this class has sent e-mail to exactly two other students in this class." 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 "Sent(x,y)" which states that x sent an email to y. The statement says that for every student x in this class, for every student y in this class, if x sent an email to y, then x did not sent exactly two emails to y (∀ x (Student(x) -> ∀ y (Student(y) -> (Sent(x,y) → Sent(x,y) ≠ 2)))).
The negation of this statement is "There exists at least one student in this class who sent e-mail to exactly two other students in this class." 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, "Student(y)" which states that y is a student in this class and "Sent(x,y) = 2" which states that x sent exactly two emails to y. The negation says that there exists at least one student x in this class, and at least one student y in this class, such that x sent exactly two emails to y (∃ x (Student(x) & ∃ y (Student(y) & Sent(x,y) = 2))).
d) The original statement is "Some student has solved every exercise in this book." This statement can be expressed using the quantifier "there exists" (∃) and the predicates "Student(x)" which states that x is a student, "Exercise(y)" which states that y is an exercise in this book, and "Solved(x,y)" which states that x has solved exercise y. The statement says that there exists at least one student x such that for every exercise y in this book, x has solved y (∃ x (Student(x) & ∀ y (Exercise(y) -> Solved(x,y)) )).
The negation of this statement is "For every student, there exists at least one exercise in this book that the student has not solved." This negation can be expressed by using the quantifier "for all" (∀) and the predicates "Student(x)" which states that x is a student, "Exercise(y)" which states that y is an exercise in this book, and "!Solved(x,y)" which states that x has not solved exercise y. The negation says that for every student x, there exists at least one exercise y in this book such that x has not solved y (∀ x (Student(x) -> ∃ y (Exercise(y) & !Solved(x,y)))).
e) The original statement is "No student has solved at least one exercise in every section of this book." This statement can be expressed using the quantifier "for all" (∀) and the predicates "Student(x)" which states that x is a student, "Section(y)" which states that y is a section of this book, "Exercise(z)" which states that z is an exercise, "Solved(x,z)" which states that x has solved exercise z and "In(z,y)" which states that exercise z is in section y. The statement says that for every student x, for every section y of this book, there does not exist an exercise z such that x has solved z and z is in y (∀ x (Student(x) -> ∀ y (Section(y) -> !(∃ z (Exercise(z) & Solved(x,z) & In(z,y))))).
The negation of this statement is "There exists at least one student and at least one section in this book, such that the student has solved at least one exercise in that section." This negation can be expressed by using the quantifier "there exists" (∃) and the predicates "Student(x)" which states that x is a student, "Section(y)" which states that y is a section of this book, "Exercise(z)" which states that z is an exercise, "Solved(x,z)" which states that x has solved exercise z and "In(z,y)" which states that exercise z is in section y. The negation says that there exists at least one student x and at least one section y in this book such that there exists an exercise z such that x has solved z and z is in y (∃ x (Student(x) & ∃ y (Section(y) & ∃ z (Exercise(z) & Solved(x,z) & In(z,y)))).
