Final answer:
To have a relation with the domain as a set and the range as a proper class, select V, the range, to be a proper class like the class of all sets, while limiting the domain to a specific set, like the natural numbers ℕ.
Step-by-step explanation:
To modify the relation ∈, which is defined as {(x, y) : x ∈ y ∧ x ∈ U ∧ y ∈ V} where U is the universe, such that the domain is a set but the range is a proper class, we can adjust the definition of V. We need to choose V such that it is not a set but a proper class, for example, the class of all sets. This would ensure that the range of our relation can include elements that are not part of any set (since a proper class contains elements that cannot be gathered into a single set). The domain, however, can be a specific set within U, such as the set of all natural numbers, ℕ. Our revised relation would thus be {(n, y) : n ∈ ℕ ∧ y ∈ V}, where the domain is the set of natural numbers ℕ (a set), and the range is V (a proper class).