7. (1.4 Predicates and Quantifiers) Let P(x) be the statement "x is a professor," let Q(x) be the statement "x is ignorant," and let R(x) be the statement "x is vain." Select the correct translation of the expression (x)(P(x) = -R(x)) where the domain consists of all people. a. No professors are ignorant. b. All ignorant people are vain. c. No professors are vain. d. All ignorant professors are vain. 8. (1.5 Nested Quantifiers) Let S(x) be the statement "x is a student," F(x) the predicate "x is a faculty member," and A(x,y) the predicate "x has asked y a question, "where the domain consists of all people associated with your school. Select the expression that is the translation of the statement "Some student has never been asked a question by a faculty member." a. 3x(F(x) A Vy(S() - -A(y,x))) b. 3x(S(x) A Vy(F(y) A(y,x))) C. Vy(F(y) 3x(S(x) A(x,y))) d. 3x(F(x) A Vy((F(y) (y + x) - A(x,y)) 9. (1.5 Nested Quantifiers) A discrete mathematics class contains 1 mathematics major who is a freshman, 12 mathematics majors who are sophomores, 15 computer science majors what are sophomores, 2 mathematics majors who are juniors, 2 computer science majors who are juniors, an 1 computer science major who is a senior. Let P(s.c, m) be the statement that students has class standing c and is majoring in m. The variable s ranges over students in the class, the variable c ranges over the four class standings, and the variable m ranges over all possible majors. Which of the following propositions is true. a. Esacam(PCs,c,m) (c + junior) (m # mathematics)) b. Vs(EcP(s.c, computer science) VamP(s, sophomore, m)) c. VsacP(s.c, computer science) d. 3mVcsP(s,c,m)