Reference no: EM13320710
1. State whether the following statements are true or not. You must give a BRIEF explanation or show a counter example to receive full credit.
(a) If a subset of a regular language is regular then all of its subsets are regular.
(b) (0111 + 1011 + 1101 + 1110)* is the regular expression for the set of all strings with 3n0 = n1 where n0 and n1 respectively denote the numbers of 0s and 1s in w.
(c) Let R1 = aba+ and R2 = ab+a be two regular expressions then R1 ∪ R2 = ab(a + b)*b.
(d) With pumping lemma, we can prove that L = {w : w = 1000} is a regular language.
2. The following table shows a non-associative operation closed under set A = {a, b, c, d} The left-to-right and right-to-left evaluation values of a string w, when interpreted as an expression, are represented as wlr and wrl, respectively. As an example if w = adb then wlr = (ad)b = ab = c and wrl = a(db) = ac = b.
(a) Give a DFA for the following language with the alphabet A, the first input to the DFA will be the left-most letter of the string:
L = {ω : ω ∈ A*, ωlr = d }
(b) Give an NFA with minimum number of states for the following language with the alphabet A, the first input to the NFA will be the left-most letter of the string:
L = {ω : ω ∈ A*, ωrl = d }
3. Give a regular expression for the set of all strings from f0, 1g , which are base 2 represen-tations of a number which is a multiple of 5. For example, the string 1111 which represents (1111)2 and equals 15, must be generated by the regular expression since it is a multiple of 5.
4. If L is a regular language prove that the following languages are also regular.
5. Prove or disprove that the following languages are regular.
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