Please answer all questions.

Unless otherwise noted, the alphabet for all questions below is assumed to be Sigma = {0, 1}. This question develops your ability to devise regular expressions, given an explicit definition of a language. For each of the following languages, prove they are regular by giving a regular expression which describes them. Justify your answers. L = {x | x begins with a 0 and ends with a 1}. L = {x | x contains at least four 0's}. L = {1, 11, epsilon}. L = {x | the length of x is at most 3}. L = {x | x doesn't contain the substring 110}. L = {x | |x| > 0, i.e. x is non-empty}. This question tests your understanding of how to translate a regular expression into a finite automaton. Using the construction of Lemma 1.55, construct NFAs recognizing the languages described by the following regular expressions. R = 0*. R = (0 Union 1)*111(0 Union 1)*