Ken wrote:

> I've been staring at this for a day, and it all just hit me like brick and seems obvious now!

Great! It would have been clearer with more pictures. I wish some god of pictures like Michael Hong would draw a set equipped with two partitions \\(P\\) and \\(Q\\), say red and blue, and show how to get from one point \\(x\\) to another point \\(y\\) by hopping like this:

\\( x\\) and \\(z_1\\) are in the same part of \\(P\\)

and

\\(z_1\\) and \\(z_2\\) are in the same part of \\(Q\\)

and so on, maybe not too long, ending with something like

\\(z_2\\) and \\(y\\) are in the same part of \\(P\\).

For extra fun, then draw the partition \\(P \vee Q\\).

> I've been staring at this for a day, and it all just hit me like brick and seems obvious now!

Great! It would have been clearer with more pictures. I wish some god of pictures like Michael Hong would draw a set equipped with two partitions \\(P\\) and \\(Q\\), say red and blue, and show how to get from one point \\(x\\) to another point \\(y\\) by hopping like this:

\\( x\\) and \\(z_1\\) are in the same part of \\(P\\)

and

\\(z_1\\) and \\(z_2\\) are in the same part of \\(Q\\)

and so on, maybe not too long, ending with something like

\\(z_2\\) and \\(y\\) are in the same part of \\(P\\).

For extra fun, then draw the partition \\(P \vee Q\\).