Question 83
For any finite sets $X$ and $Y$ of positive integers, denote by $f_{X}(k)$ the $k^{\text{th}}$ smallest positive integer not in $X$, and let
$$
X * Y = X \cup \{ f_{X}(y) : y \in Y \} .
$$
Let $A$ be a set of $a \gt 0$ positive integers, and let $B$ be a set of $b \gt 0$ positive integers. Prove that if $A * B = B * A$, then
$$
\begin{equation*}
\underbrace{A *(A * \cdots *(A *(A * A)) \ldots)}_{A \text{ appears } b \text{ times }} = \underbrace{B *(B * \cdots *(B *(B * B)) \ldots)}_{B \text{ appears } a \text{ times }} . \tag{U.S.A.}
\end{equation*}
$$
$$
X * Y = X \cup \{ f_{X}(y) : y \in Y \} .
$$
Let $A$ be a set of $a \gt 0$ positive integers, and let $B$ be a set of $b \gt 0$ positive integers. Prove that if $A * B = B * A$, then
$$
\begin{equation*}
\underbrace{A *(A * \cdots *(A *(A * A)) \ldots)}_{A \text{ appears } b \text{ times }} = \underbrace{B *(B * \cdots *(B *(B * B)) \ldots)}_{B \text{ appears } a \text{ times }} . \tag{U.S.A.}
\end{equation*}
$$