SCHWEITZER 2002 Problem 3

Put $\mathbb{A}=\{ \mathrm{yes}, \mathrm{no} \}$ . A function $f\colon \mathbb{A}^n\rightarrow \mathbb{A}$ is called a decision function if (a) the value of the function changes if we change all of its arguments; and (b) the values does not change if we replace any of the arguments by the function value. A function $d\colon \mathbb{A}^n \rightarrow \mathbb{A}$ is called a dictatoric function, if there is an index $i$ such that the value of the function equals its $i$ th argument. The democratic function is the function $m\colon \mathbb{A}^3 \rightarrow \mathbb{A}$ that outputs the majority of its arguments. Prove that any decision function is a composition of dictatoric and democratic functions.