(a) Let the domain of discourse consist of all persons. Let C(x) be the predicate "x is a child," and M(x) be the predicate "x likes mathematics," and B(x) be the predicate "x owns a bike." Translate each of the following statements into logical expressions using predicates, quantifiers, and logical connectives.
(i) There is a person who owns a bike.
(ii)There is a child who owns a bike.
(iii) Every child likes mathematics.

(b) (4 points] Let the domain of discourse consist of all students, for both variables x and y. Let L(x, y) be the predicate stating that "x has listened to y." Use quantifiers and logical connectives to express each of the following statements. (i) Bill has listened to Kate.
(ii) There exists a student who has listened to Peter.
(iii) Frank has listened to some student. (iv) Every student has listened to Mary.

(e) 3 points) Let the domain of discourse consist of all animals. Let D(x) be the predicate "x is a dog" and B(x) the predicate "x can bark."
(i) Translate the statement "Every dog can bark" into a logical expression using predicates, quantifiers, and logical connectives.
(ii) Form the negation of the logical expression in your answer in (i).
(iii) Express your answer in (ii) equivalently in such a way that no negation symbol is to the left of a quantifier