Question 15
For any finite sets $X$ and $Y$ of positive integers, denote by $f_{X}(k)$ the $k^{\mathrm{th}}$ smallest positive integer not in $X$, and let
$$
X * Y=X \cup\left\{f_{X}(y): y \in Y\right\}
$$
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
$$
\underbrace{A *(A * \cdots *(A *(A * A)) \ldots)}_{A \text { appears } b \text { times }}=\underbrace{B *(B * \cdots *(B *(B * B)) \ldots)}_{B \text { appears } a \text { times }} .
$$
$$
X * Y=X \cup\left\{f_{X}(y): y \in Y\right\}
$$
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
$$
\underbrace{A *(A * \cdots *(A *(A * A)) \ldots)}_{A \text { appears } b \text { times }}=\underbrace{B *(B * \cdots *(B *(B * B)) \ldots)}_{B \text { appears } a \text { times }} .
$$