For a string w, recall that wR denotes the reverse of w, i. e., the string obtained by reversing the order of the symbols of w. For a language L, define S (L) = {w∈L:wR∈L}. If L is regular, is S (L) also regular? Prove your claim.

One arrangement is recursively (or inductively) characterize a turning around procedure on regular articulations, and apply that procedure on the regular articulation for L. Specifically, given a regular articulation R, reverse (R) is

• a for some a ∈ Σ,

• Σ if R = Σ,

• ∅ if R = ∅,

• (reverse (R1) ∪ reverse (R2)), if R = R1 ∪ R2,

• (reverse (R2) ◦ reverse (R1)) if R = R1◦ R2, or

• (reverse (R1) ∗), if R = (R∗1)