Question 44
We say that two non-negative integers are related if their sum uses only the digits $0$ and $1$. For example, $22$ and $79$ are related. Let $A$ and $B$ be two infinite sets of non-negative integers such that:
(1) if $a \in A$ and $b \in B$, then $a$ and $b$ are related,
(2) if $c$ is related to every member of $A$, then it belongs to $B$,
(3) if $c$ is related to every member of $B$, then it belongs to $A$.
Show that in one of the sets $A$, $B$ we can find an infinite number of pairs of consecutive numbers.
(1) if $a \in A$ and $b \in B$, then $a$ and $b$ are related,
(2) if $c$ is related to every member of $A$, then it belongs to $B$,
(3) if $c$ is related to every member of $B$, then it belongs to $A$.
Show that in one of the sets $A$, $B$ we can find an infinite number of pairs of consecutive numbers.