Looking for a formal mathematical proof by induction preferably. Step by Step if possible for each question.

Problem 2 . Prove that for any singleton string and for any string , (1) ()=. (2) ()r=. (3) r=. (4) = Problem 3. Prove that for any string = and for any string , (1) ()=. (2) ()r=r. Problem 4. Prove that for any non-empty string T, (1) ()=. (2) ()r=r