To prove the number of functions from set X to set Y, where n(X) = p and n(Y) = q, the key is to consider each element of X and its possible images in Y. Each element in X has q choices for its image in Y, leading to a total of q^p combinations when considering all p elements in X. The initial confusion stemmed from misinterpreting the relationship between functions and relations, but recognizing that each element independently contributes to the total count clarifies the result. Therefore, the number of functions from X to Y is indeed q^p. This understanding simplifies the proof significantly.