A Note on Asymmetry of Set Functions

This short note presents a formal combinatorial result on functions defined over all subsets of a finite set. We consider functions f: P(S) → R satisfying the additive propertyf(A ∪ B) + f(A ∩ B) = f(A) + f(B) for all subsets A,B ⊆ S.The main theorem proves that any such function must be an affine combination of the indicator functions of singletons plus a constant term:f(A) = c_0 + Σ_{i∈A} c_i. The note includes a fully formal proof using induction, highlights the non-obvious structural rigidity imposed by the additive condition, and provides references to classical literature on functional equations and set functions. This publication is intended as a concise, self-contained mathematical note suitable for dissemination as a technical reference, educational material, or a formal short note in combinatorics. The PDF is included, formatted with clear theorem and proof blocks for readability.