1. Consider the operations ProperSuffix on languages over the alphabet ={a,b,c} ProperSuffix(L)={yw is in L,x is not lambda and w=xy} SomeHalf (L)={y there exists a string x,x=y and either xy is in L or yx is in L} a.) \& b.) Show Regular Languages are closed under each of these operations. ProperSuffix is easy if you consider the meta technique we discussed for closures like prefix, substring, and suffix. Hints: To see techniques for this, look at sample \#1a located in Also, look in the notes (COT6410FormalLanguages AutomataTheory.pdf) on pages 75-78 SomeHalf is only slightly more challenging as it 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. 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 as I did mine can help with partial credit if you have an error in your states or transition function. Hints: The best place to look for a similar example is in sample \#1b located ir Also, look in the notes (COT6410FormalLanguages AutomataTheory.pdf ) on pages 67-72 as this might help you understand the general idea but the Half sample is the better source