We know that if there are two languages, L1 and L2, if L1 and L2 are regular, the intersection of those two is also regular. Suppose we have two machines, M1 and M2, and using them, a new machine M3 is built. If M1 had k>0 states and M2 had l>0 states, the new machine has to have kl states at minimum.
How would I go about trying to prove this? I've been stuck on this for quite some time. I know that we would have to construct the languages L1 and L2 so that M1 and M2 accept them, respectively, which in turn allows the smallest machine accepts L1 intersects L2 has k x l states.
Any help would be greatly appreciated.