For game #2, first get a handle on the conditions:

#1: P

_{U}; ~O

_{N}#2: P

_{N} and P

_{U} --> S

_{N} and S

_{U}; ~S

_{N} or ~S

_{U} --> ~P

_{N} or ~P

_{U} (contrapositive)

#3: J

_{N} and J

_{U} --> ~R

_{N} and ~R

_{U}; R

_{N} or R

_{U} --> ~J

_{N} or ~J

_{U} (contrapositive)

#4: ~J

_{N} and ~J

_{U} --> P

_{N}; ~P

_{N} --> J

_{N} or J

_{U} (contrapositive)

#5: R

_{N} or R

_{U} --> ~S

_{N} and ~S

_{U}; S

_{N} or S

_{U} --> ~R

_{N} and ~R

_{U} (contrapositive)

A couple of the conditions can be simplified:

J

_{N} and J

_{U} <--|--> R (#3)

R <--|--> S (#5)

Combining the first two conditions, we know that if P

_{N} is on sale, then S

_{N} and S

_{U} are on sale. We can also link the fourth condition into the equation:

~J

_{N} and ~J

_{U} --> P

_{N} --> S

_{N} and S

_{U}Here are explanations for a couple of the questions:

10. Starting with the first condition given in choice D, due to the fourth rule, new pop must be on sale. Since both types of pop are on sale, both types of soul must be on sale (second rule), which contradicts the second condition given in answer choice D.

11. From the fourth rule, we know that new pop is on sale. Since used pop is on sale (first rule), the second rule is triggered, and both types of soul must be on sale. Since rap and soul cannot both be on sale (fifth rule), we know that rap is not on sale. Thus, we can eliminate B, C, D, and E, and A is left as the correct answer. We cannot infer anything about used opera.

Here's a diagram for the fourth game:

PrepTest 31, Game 4