Here's another way to think about this, and Boolean Algebra. In my opinion, it's much more intuitive and much more useful. Forget about Boolean unionization "OR" as "addition" and Boolean intersection "AND" as "multiplication". Unfortunately, some book and perhaps even your teacher probably tried to explain it to you in terms of some multiplication and addition, and this interpretation can actually get traced back to Boole himself, but I digress. Sure, Boolean intersection behaves just like ordinary multiplication on {0, 1}, but, as I feel sure you know, Boolean unionization simply doesn't behave like ordinary addition on {0, 1}, since $ (1 \lor 1)=1 $. So, keep that in mind and that (1+1)=2 if you perhaps revert back to thinking about Boolean algebra in terms of ordinary addition and multiplication. But, of course, I suspect you want some way to think about Boolean Algebra concretely... so here goes...
Think of Boolean intersection as taking the minimum function, and think of Boolean unionization as taking the maximum function. The minimum function behaves exactly like Boolean intersection on {0, 1}, and the maximum function behaves exactly like Boolean unionization on {0, 1}. Now, you simply can't cancel things out here under this interpretation either, and it's much simpler than thinking in terms of multiplication or addition.