A DFA is said to be minimal if there is no DFA with fewer states that accepts the same language.

The product construction yields a DFA that accepts the union of two regular languages. Sometimes the construction gives a minimal DFA, but it often gives a DFA with more states than necessary. Show that even if we start with two minimal DFAs, the resulting DFA may not be minimal.

More specifically: Give an example of two regular languages A, B such that if we take two minimal DFAs MA, MB that accept A and B, respectively, and construct M using the product construction to accept A B, then M is not a minimal DFA. In your solution define A and B, show transition diagrams for MA, MB, and M, and give a DFA M0 for A B with fewer states than M.