If (f:A\to B) with (A=\varnothing) and (B={1,2}), how many functions exist from (A) to (B)?

Correct answer: (1)

The direct answer is B: exactly one function exists. A function must assign one output to each element of its domain. When the domain is empty, there are no elements that need assignments, so the empty assignment satisfies the function rule automatically. There is also only one empty set of ordered pairs, namely the empty function. The counting formula confirms this: the number of functions from a set of size \(|A|\) to a set of size \(|B|\) is \(|B|^{|A|}\), so here it is \(2^0=1\). Option A, 0, is wrong because the empty function does exist. Option B, 1, is correct. Option C, 2, incorrectly treats the two codomain elements as two choices even though there is no domain input to receive a choice. Option D, 4, would be \(2^2\), relevant to two domain elements, not an empty domain. The important idea is that there is exactly one way to make no assignments.