 Solution

The standard relational guarded fragment contains $\boxed{\text{(i) and (ii)}}$. In (i), $Student(x)$ guards the only quantified variable:The constants in the course-taking relation do not introduce additional free variables that need guarding.

For (ii), use the two implications from part (c)(i). Each outer universal quantification is guarded by its unary category predicate. The inner existential quantification can be writtenwhose guard contains both $x$ and $y$. Sentence (iii) uses a nonconstant function and so is outside the standard relational guarded fragment. This conclusion uses the original vocabulary; it does not silently replace the counting term by a fresh relation.