Disjoint-set union operation ($\mathrm{unite}(x,y)$)

The disjoint-set union operation first uses the disjoint-set find operation to locate the blocks of a set partition containing $x,y$. If they are different blocks of a set partition $A,B$, replace them by their union $A\cup B$; every other block of a set partition stays as it was. If the two disjoint-set representatives agree, no merge is needed. Here unite returns True exactly when it merges two different blocks of a set partition. A successful call decreases their count by one; a redundant call preserves it. The name union is also customary. Some interfaces instead accept already found disjoint-set representatives, making that linking step a separate operation.

For an original example, initialize $X=\{1,2,3,4,5,6,7\}$ as seven singleton sets. Apply the calls in this order:The first five calls succeed. After the third, the partition of a set is $\{\{1,2,3,4\},\{5\},\{6\},\{7\}\}$; after the fifth it is $\{\{1,2,3,4,5,6\},\{7\}\}$. The sixth returns False: $1,2$ are already together. The blocks of a set partition number $7,6,5,4,3,2,2$, counting the initial state. A disjoint-set find operation on $7$ cannot merge it with any other member.

This is exactly the kind of changing membership needed by Kruskal component merging: an accepted edge joins two current connected components of a graph, while an edge whose endpoints already have equal disjoint-set representatives is rejected. The existing Kruskal's algorithm worked example follows that application. A disjoint-set forest stores membership efficiently without repeatedly enumerating every block of a set partition.

When merging two blocks in quick-find, let $a=re{p}_0(x)$ and $b=re{p}_0(y)$ be fixed saved labels. Replacing every old occurrence of $b$ by $a$ preserves all other blocks. A scan loop invariant says processed cells already have their specified final labels while unprocessed cells still have their old labels. Recomputing $rep(y)$ during this mutation can change the replacement label after the cell for $y$ is processed, causing remaining old-$b$ cells to be missed. Saving both labels makes the update criterion stable and the invariant provable.