Question

If L is a regular language, prove that the language {uv : u ∈L, v ∈L^R} is also regular

Answer 1

Given L is regular and L^' = {uv,u elementof L , v elementof L^R} if we can able to built an finite automation that recognizes L^' then L^' is also regular . Now, L is regular and M be the finite automata that recognizes L rightarrow finite automata for L^R can be generated from M, by rightarrow reversing arcs of M rightarrow setting initial state in M to final state & connect all final states with epsilon transition to a state and set it as initial state. Rightarrow This would be M^R rightarrow Now connect M & M^R with epsilon transition U elementof M & V elementof M^R rightarrow This machine recognize L^' rightarrow Since we have built finite automata for L^' , it is a regular language.