Statements: • Some cats are dogs. • All dogs are black. Conclusions: I. Some cats are black. II. Some dogs are black. Choose the option that must follow.

Introduction / Context:
Combining a particular overlap with a universal property often yields two distinct necessary consequences: one about the particular elements involved and one about the whole class under the universal statement.

Given Data / Assumptions:

• Some Cats are Dogs (∃x ∈ Cats ∩ Dogs).
• All Dogs are Black (Dogs ⊆ Black).

Concept / Approach:
Take the witnessed cat–dog individual; because all dogs are black, that same individual is black. Therefore “some cats are black.” Separately, “all dogs are black” implies “some dogs are black,” provided at least one dog exists— and existence is guaranteed by the first premise.

Step-by-Step Solution:
1) From “Some Cats are Dogs,” pick witness c1 ∈ Cats ∩ Dogs.2) Since Dogs ⊆ Black, c1 ∈ Black, yielding “Some Cats are Black” ⇒ Conclusion I true.3) The existence of c1 establishes that at least one dog exists. With Dogs ⊆ Black, it follows that “Some Dogs are Black” ⇒ Conclusion II true.

Verification / Alternative check:
Model: Dogs = {d1}, Cats = {d1, c2}, Black = universal set. Premises hold. Both conclusions hold, witnessed by d1 (who is both a cat and a black dog).

Why Other Options Are Wrong:
Any option denying I or II contradicts direct, necessary consequences of the premises. “Data inadequate” is incorrect because the first premise supplies the needed existence.

Common Pitfalls:
Forgetting to use the existence guaranteed by “some” to justify that at least one dog (hence a black dog) exists.

Final Answer:
Both Conclusions I and II follow.