Answer
a. True. Every student chose at least one dessert: Uta chose pie, Tim chose both pie and cake, and Yuen chose pie.
b. False. Yuen did not choose a salad.
c. This statement says that some particular dessert was chosen by every student. This is true: Every student chose pie.
d. This statement says that some particular beverage was chosen by every student. This is false. Uta and Tim chose milk, but Yuen chose soda.
e. This statement says there is a particular item that was not chosen by everyone. This is true. Fish is chosen by Tim and Yuen but not Uta.
f. This statement says there is a particular station Z from which all students chose a particular item I. This is true because Uta, Tim, and Yuen all chose from the station Dessert the item pie.
Work Step by Step
Recall if you want to establish the truth of a statement of the form $\forall x \in D, \exists y \in E$ such that P(x,y), your challenge is to allow someone else to pick whatever element x in D they wish and then you must find an element y in E that "works" for that particular x.
If you want to establish the truth of a statement of the form $\exists x \in D$ such that $\forall y \in E$, P(x,y), your job is to find one particular x in D that will "work" no matter what y in E anyone might choose to challenge you with.
