Please enter the following multiple choice and give explanation for each answer.

1. Which of these statements is a correct translation of the statement Every student in this class knows the Java programming language where C(x)is the statement x is a student in this class and J(x) is the statement x know Java. Assume that the universe of discourse is the set of all people.

a. x (C(x) J(x))

b. x (C(x) J(x))

c. x (C(x) J(x))

d. x (C(x) J(x))

e. x (C(x) J(x))

2. Suppose P(x,y) is a predicate where the universe for x and y is [1,2,3]. Also suppose that the predicate is true in the following cases P(1,2), P(2,1), P(2,2), P(2,3), P(3,1), P(3,2) and false otherwise. Determine which of the following statements is FALSE

a. y x P(x,y)

b. y x P(x,y)

c. x y (x y P(x,y))

d. x y P(x,y)

3. Express the following statements in symbols. Some computer science majors is taking no courses. Using the following: C(x) is the statement x is a computer science major, T(x,y) is the statement x is taking y with the universe for x being the set of all students, and the universe for y being the set of all courses.

a. x y (C(x) T(x,y))

b. y x (C(x) T(x,y))

c. x y (C(x) T(x,y))

d. x y (T(x,y) C(x))

4. Suppose P(x,y,z) is a predicate where the universe for x,y, and z is [1,2]. Also suppose that the predicate is true in the following cases: P(1,1,1), P(1,1,2), P(2,1,1), P(2,2,2) and false otherwise. Determine which of the following quantified statements is TRUE.

a. x y z P(x,y,z)

b. x z y P(x,y,z)

c. x y z P(x,y,z)

d. x z y P(x,y,z)