Q.Classify each of the following as empty, singleton, finite, or universal (taking $\mathbb{N}$ as the universal set for this problem):

Match each set's description against the four special-set tests: does the condition have no solution, exactly one, a countable-and-finite number, or is it "everything under discussion"?

(i) $x + 5 = 2 \Rightarrow x = -3$. Since $-3 \notin \mathbb{N}$, no natural number satisfies the condition, so the set has no elements — it is the empty set, $\emptyset$.

(ii) $\{7\}$ contains exactly one element, $7$. This is a singleton set.

(iii) $x \in \mathbb{N}, x < 6$ gives $x = 1, 2, 3, 4, 5$ — five elements, a definite finite count. This is a finite set, $\{1, 2, 3, 4, 5\}$. …