Q.Let $A = \{x, y\}$. Write the power set $P(A)$ and verify that $n(P(A)) = 2^2$.

List every possible subset of the 2-element set, remembering to include the empty set and the full set, then count them against the formula.

A 2-element set has exactly four subsets: empty, two singletons, and the full set.

Step 1 — List all subsets. Size 0: $\emptyset$. Size 1: $\{x\}$, $\{y\}$. Size 2: $\{x,y\}$ (the whole set).

Step 2 — Collect. $P(A) = \{\emptyset, \{x\}, \{y\}, \{x,y\}\}$ — 4 subsets total.

Step 3 — Verify with the formula. $n(A) = 2$, so $n(P(A)) = 2^2 = 4$, matching the direct count.

Omitting $\emptyset$ or the full set $\{x,y\}$ from the listing, under the mistaken idea that a power set only lists "proper" subsets.