A recursive definition for the language EVENnotAAA over the alphabet $=\{$a b$\}$ must be compiled

where EVENnotAAA consists of all words of even length and not containing the substring aaa.

Give $($i$)$ an appropriate universal set,

$($ii$)$ the generator$($s$)$ of EVENnotAAA, and

$($iii$)$ an appropriate function on the universal set, and then

$($iv$)$ use these concepts to write down a recursive definition of the language EVENnotAAA