Let A be a finite set of size n. The number of elements in the power set of A…

For a finite set S with k elements, the power set P(S) has exactly 2k elements, since each of the k elements can independently be included in or excluded from a subset. For finite sets A and B, the Cartesian product A × B has |A| × |B| ordered pairs, since every element of A pairs with every element of B.

1. Since |A| = n, the Cartesian product A × A has |A| × |A| = n × n = n2 ordered pairs.

2. A × A is itself a set with n2 elements, so by the power-set rule its power set has 2(n²) elements.

3. This matches exactly: the power set of A × A has 2(n²) elements.

Cross-check with n = 2: let A = {a, b}. Then A × A has 2 × 2 = 4 ordered pairs, and its power set has 24 = 16 subsets. Since n2 = 4 here, this equals the power set of A × A for n = 2, confirming the formula.