Union-Find with link-by-rank to represent a binary field with simple operations
In each node store an additional field named "value". The value of a root $r$ represents $\text{FIND}(r)$ (so the root itself is not necessarily $\text{FIND}(r)$). For $\text{SET_TO_1}(i)$, let $r$ be the root of the tree $i+1$ belongs to, we in addition change the value of the new root to the value of $r$. Now which root becomes the new root does not matter at all.