Automata Theory problemWill Rate Immediately

(15 pt) For every two languages L1, L2 C I* and the reverse of a language L be defined as LR { * | w w = 2R for some x L}. Then, prove that (LL2) R = L5LR. (15 pt) Let = {a,b} be an alphabet. Design a DFA that accepts strings whose parts including only a's is of even length. For instance, strings aa and baabaaaa are accepted whereas strings aba and aaab are rejected. (15 pt) For every two languages L1, L2 C I* and the reverse of a language L be defined as LR { * | w w = 2R for some x L}. Then, prove that (LL2) R = L5LR. (15 pt) Let = {a,b} be an alphabet. Design a DFA that accepts strings whose parts including only a's is of even length. For instance, strings aa and baabaaaa are accepted whereas strings aba and aaab are rejected