Let D be the set of all students at your school, and let M(s) be "s is a math major," let C(s) be "s is a computer science student," and let E(s) be "s is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates M(s), C(s), and E(s). (a) There is an engineering student who is a math major. ∀ s D, E(s) ∧ ~M(s) ∀ s D, ~E(s) ∧ ~M(s) ∃ s D such that ~E(s) ∧ ~M(s) ∃ s D such that E(s) ∧ M(s) ∃ s D such that E(s) ∧ ~M(s) ∀ s D, E(s) ∧ M(s) (b) Every computer science student is an engineering student. ∃ s D such that ~C(s) → E(s) ∀ s D, ~C(s) → ~E(s) ∀ s D, E(s) → C(s) ∃ s D such that E(s) → C(s) ∃ s D such that C(s) → E(s) ∀ s D, C(s) → E(s) (c) No computer science students are engineering students. ∃ s D such that ~C(s) → E(s) ∃ s D such that C(s) → ~E(s) ∀ s D, ~C(s) → E(s) ∀ s D, ~E(s) → C(s) ∃ s D such that E(s) → C(s) ∀ s D, C(s) → ~E(s) (d) Some computer science students are also math majors. ∀ s D, C(s) → M(s) ∃ s D such that C(s) ∧ M(s) ∃ s D such that C(s) ∧ ~M(s) ∀ s D C(s) ∧ ~M(s) ∀ s D, C(s) ∧ M(s) ∃ s D such that C(s) ∧ M(s) ∧ E(s) (e) Some computer science students are engineering students and some are not. (∃ s D such that C(s) ∧ E(s)) ∧ (∃ s D such that E(s) ∧ ~E(s)) ∀ s D, C(s) → (E(s) ∧ ~E(s)) (∀ s D, C(s) ∧ E(s)) ∧ (∀ s D, C(s) ∧ ~E(s)) (∃ s D such that C(s) ∧ E(s)) ∧ (∃ s D such that C(s) ∧ ~E(s)) ∀ s D, (E(s) → C(s)) ∧ (~E(s) → ~C(s)) ∃ s D such that C(s) → (E(s) ∧ ~E(s))