The number of binary relations on a set with n elements is:

A binary relation on a set A with n elements is defined as any subset of the Cartesian product A × A. First, determine the size of this Cartesian product. Since set A has n elements, the product A × A contains exactly n × n = n² ordered pairs. A binary relation is simply a selection of some or all of these possible pairs, meaning it corresponds to any subset of A × A. The total number of distinct subsets of a set containing k elements is given by 2^k. Here, the 'set' we are taking subsets of is A × A, which has k = n² elements. Therefore, the total number of possible binary relations is 2 raised to the power of n², or 2^(n²). Option A (n²) represents only the number of possible pairs, not the subsets. Option B (2^n) would be the number of relations on a set if we were choosing subsets of A itself, not pairs. Thus, the correct count is 2^(n²).