Cross Product Method over DFAs

We will try to understand this property using one example.

So as you see in the above picture, we have taken two DFAs
  1. Even no of a's
  2. Even no of b's
Languages represented by them are respectively
  1. L1 = {ε, baa, aa, aba, aab, aaaa, ... }
  2. L2 = {ε bb, abb, bab, bba, ...}

After taking cross product we will find the below DFA, already we have seen this DFA in previous exampels
As, L = {ab, aab, abb, aaab, ...}

How we have designed:

{A, B} X {C, D} = {AC, AD, BC, BD}
We should know the transition of the above four states

Note: clearly the language is infinite
You can also take some examples from the previous exercises for practice.