please help me on 4 Thanks Define sR to be the reversal of

please help me on #4 . Thanks

Define s^R to be the \"reversal\" of the string .s, i.e. the symbols written in reverse order. For example. ([]])^R =]][Recall B was the set of brackets created with our recursive definition. Prove that V_s subset B: s^R subset B Recall our recursive definition of the set S: Base case lambda subset S. Constructor case If x subset S, then axa subset S and bxb subset S. Using the same definition of reversal from the previous problem, prove V_x subset S: x^R = x

Solution

S

We start from this ans construct the set by the rules axa S and bxbS

Now if we have a string p, which does not conform to this rule. So either the start and end of p should a,b respectievly or b,a. Then p = aqb or bqa. But if we take any string from the set S we cannot construct p, because have only two and this does not conform to both of them.

Hence p does not belong to S. So for all string reversal of the string is same as the string.