Question 35
For any two finite sets $X$ and $Y$ of positive integers, define:
(1) $f_X(1) = \min\{n : n \in \mathbb{N}, n \notin X\}$, where $\mathbb{N}$ denotes the set of all positive integers;
(2) $\forall k \gt 1$, $f_X(k) = \min\{n : n \gt f_X(k-1), n \notin X\}$;
(3) $X \star Y = X \cup \{f_X(y) : y \in Y\}$.
Let $a, b$ be positive integers, $A$ a set of $a$ positive integers, and $B$ a set of $b$ positive integers.
Prove: If $A \star B = B \star A$, then
$$
A \star (A \star \cdots \star (A \star (A \star A)) \cdots) = B \star (B \star \cdots \star (B \star (B \star B)) \cdots),
$$
where the left side has $b$ $A$'s and the right side has $a$ $B$'s.
(1) $f_X(1) = \min\{n : n \in \mathbb{N}, n \notin X\}$, where $\mathbb{N}$ denotes the set of all positive integers;
(2) $\forall k \gt 1$, $f_X(k) = \min\{n : n \gt f_X(k-1), n \notin X\}$;
(3) $X \star Y = X \cup \{f_X(y) : y \in Y\}$.
Let $a, b$ be positive integers, $A$ a set of $a$ positive integers, and $B$ a set of $b$ positive integers.
Prove: If $A \star B = B \star A$, then
$$
A \star (A \star \cdots \star (A \star (A \star A)) \cdots) = B \star (B \star \cdots \star (B \star (B \star B)) \cdots),
$$
where the left side has $b$ $A$'s and the right side has $a$ $B$'s.