I've been thinking about this notion that maps which don't preserve operations lose information:

> There is no operation on the booleans \\( true, false \\) that will always follow suit with the joining of systems: our observation is inherently lossy

Any map from more than two elements to \\( \\{false, true\\} \\) will lose information in some sense. So that's not the sense of "lossy" we're talking about here. I guess the sense in which we lose information is just this: If you apply the function before joining, you "lose information" about what you would have gotten by joining first and then applying the function.

That's the same as what it says in the chapter, but it helped me a little to put it into those words.

I also really liked that we didn't start with an operation on \\( \\{false, true\\} \\), and then ask whether it was preserved. We have a goal (impossible, in this case) of reconstructing what we would have gotten by joining first and then applying the function. Any operation that achieves this goal would be fine. This might help think about how and why we define the operations we do.

> There is no operation on the booleans \\( true, false \\) that will always follow suit with the joining of systems: our observation is inherently lossy

Any map from more than two elements to \\( \\{false, true\\} \\) will lose information in some sense. So that's not the sense of "lossy" we're talking about here. I guess the sense in which we lose information is just this: If you apply the function before joining, you "lose information" about what you would have gotten by joining first and then applying the function.

That's the same as what it says in the chapter, but it helped me a little to put it into those words.

I also really liked that we didn't start with an operation on \\( \\{false, true\\} \\), and then ask whether it was preserved. We have a goal (impossible, in this case) of reconstructing what we would have gotten by joining first and then applying the function. Any operation that achieves this goal would be fine. This might help think about how and why we define the operations we do.