$($i$)$ When asked to give a recursive definition for the language P ALINDROME

over the alphabet $\backslash$Sigma $=\{$a$,$ b$\},$ a student wrote:

Rule $1$: a and b are in P ALINDROME.

Rule $2$: If x is in P ALINDROME, then so are axa and bxb$.$

Unfortunately, all the words in the language defined above have an odd length

and so it is not all of P ALINDROME. Fix this problem.

$($ii$)$ Give a recursive definition for the language EV ENP ALINDROME of all

palindromes of even length.$($i$)$ When asked to give a recursive definition for the language P ALINDROME

over the alphabet $\backslash$Sigma $=\{$a$,$ b$\},$ a student wrote:

Rule $1$: a and b are in P ALINDROME.

Rule $2$: If x is in P ALINDROME, then so are axa and bxb$.$

Unfortunately, all the words in the language defined above have an odd length

and so it is not all of P ALINDROME. Fix this problem.

$($ii$)$ Give a recursive definition for the language EV ENP ALINDROME of all

palindromes of even length.