9. (20 pts) Determine the truth values of each of these statements if the domain for...

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9. (20 pts) Determine the truth values of each of these statements if the domain for all variables consists of all integers. Justify your answer. a. \xy(x + y < x y) b. VxVy(x+y>x-y) c. xy(x > y) d. Vy3x(x > y) 10. (10 pts) Let T(x, y) stand for the statement "Student x is taking class y." Let the domain of x be all LUO students and the domain of y be all classes offered by LUO. Express the statement below as an English sentence. ((xy) (T(x,y) T(y,z))) 6. (15 pts) Express the negation of this statement so that all negation symbols immediately precede predicates. (That is, so that no negation is outside a quantifier or an expression involving logical connectives.) Vxyz(P(x,y) Q(x, z)) 7. (15 pts) Is the compound statement p ^ (p q)) ^ (q ^r) satisfiable? Support your answer. 8. (20 pts) Use the rules of inference to show that if 3x(A(x)^B(x)), 3x(C(x) B(x)) and Vx(C(x) V D(x)) are true, then 3x(D(x) ^ B(x)) is also true, where the domains of all the quantifiers are the same. 9. (20 pts) Determine the truth values of each of these statements if the domain for all variables consists of all integers. Justify your answer. a. \xy(x + y < x y) b. VxVy(x+y>x-y) c. xy(x > y) d. Vy3x(x > y) 10. (10 pts) Let T(x, y) stand for the statement "Student x is taking class y." Let the domain of x be all LUO students and the domain of y be all classes offered by LUO. Express the statement below as an English sentence. ((xy) (T(x,y) T(y,z))) 6. (15 pts) Express the negation of this statement so that all negation symbols immediately precede predicates. (That is, so that no negation is outside a quantifier or an expression involving logical connectives.) Vxyz(P(x,y) Q(x, z)) 7. (15 pts) Is the compound statement p ^ (p q)) ^ (q ^r) satisfiable? Support your answer. 8. (20 pts) Use the rules of inference to show that if 3x(A(x)^B(x)), 3x(C(x) B(x)) and Vx(C(x) V D(x)) are true, then 3x(D(x) ^ B(x)) is also true, where the domains of all the quantifiers are the same.