For each language, either prove the language is regular (by creating an implementation with your choice of a DFA, NFA, RegEx, Regular Grammar, or using closure properties), or prove that it is not regular (by either using the pumping lemma or closure properties).

(a) {ain | n > 1} U{abn+2 | n > 1} (c) {uxvx | U, V, x {a,b}+} where {a,b}+ = {a,b}{a,b}* (d) {W E {a,b}*} : the number of occurrences of the substring ab equals the number of occur- rences of the substring ba (e) {a'bi [i, j > 0, i + j = 5}