1. Consider the operations NoLeadingOrLaggingAs over languages that include the letter 'a' in their alphabet and Third, where

NoLeadingOrLaggingAs(L) = { x | w is in L and w = a*xa* where x neither starts nor ends in an 'a' }

MidThird(L) = { y | there exists a x and z, |x| = |y| = |z| and xyz is in L }

a.) & b.) Show Regular Languages are closed under each of these operations.

NoLeadingOrLaggingAs is easy if you consider the meta technique we discussed for closures.

MidThird is challenging but can be shown using an NFA that is based on a DFA for L. Fortunately, it can be created in a reasonably obvious manneronce you understand Half, which I show as an example. You will need to describe state set, starting and accepting states, and the transition function. You do not need to show explicit examples but discussing your approach can help with partial credit if you have an error in your states or transition function