Pattern - Transitivity

If you can swap item $A$ with $B$, and swap $B$ with $C$, can you swap $A$ with $C$? Yes. You swap $A \leftrightarrow B$, then $B \leftrightarrow C$, effectively swapping $A$ and $C$.

Transitivity means swap relationships form groups. Any two items in the same group can be swapped (eventually). DSU tracks these groups perfectly.

This observation solves problems about rearranging elements through allowed swaps.