1. Consider the operations OnePairOfZerosRemoved over languages that include the letter '0' in their alphabet and LastHalf Reversed, where

OnePairOfZerosRemoved(L) = { xy | w is in L and w = x00y }

LastHalfReversed(L) = { y | there exists a string x , |x| = |y| and xy^R is in L }

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

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

LastHalfReversed is slightly 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 manner once you understand Half, which I show as an example, and you understand closure under reversal. You will need to describe the state set, starting and accepting states, and the transition function. You may not take advantage that you already know that regular languages are closed under reversal; your single construction must suffice. 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.