Question

(b) Consider the following binary relation \(\subseteq\), defined over the set of all NFAs by setting, for any NFAs \(M_1, M_2\): \(M_1 \subseteq M_2\) iff \(L(M_1) \subseteq L(M_2)\) That is, \(M_1 \subseteq M_2\) iff the language recognised by \(M_1\) is a subset of the language recognised by \(M_2\). Is \(\subseteq\) an equivalence relation? Justify your answer. [6] (c) Let A be the set of all strings over the alphabet \({0, 1}\) that do not end with more than two consecutive 0s. Examples of strings in A would be: \(\epsilon\), 1, 100, 0110. Examples of strings not in A would be: 1000, 0011000000, 11101000. (i) Is the following statement true or false?: There is a regular expression representing A You must fully justify your answer. Any definitions, theorems or propositions from the lecture notes that you use must be clearly stated in full. [5] (ii) Is it true that 11101000 \(\in A^*\)? Justify your answer. [2]

          (b) Consider the following binary relation \(\subseteq\), defined over the set of all
NFAs by setting, for any NFAs \(M_1, M_2\):

\(M_1 \subseteq M_2\) iff \(L(M_1) \subseteq L(M_2)\)

That is, \(M_1 \subseteq M_2\) iff the language recognised by \(M_1\) is a subset of the
language recognised by \(M_2\). Is \(\subseteq\) an equivalence relation? Justify your
answer.
[6]
(c) Let A be the set of all strings over the alphabet \({0, 1}\) that do not end
with more than two consecutive 0s. Examples of strings in A would
be: \(\epsilon\), 1, 100, 0110. Examples of strings not in A would be: 1000,
0011000000, 11101000.
(i) Is the following statement true or false?:
There is a regular expression representing A
You must fully justify your answer. Any definitions, theorems or
propositions from the lecture notes that you use must be clearly
stated in full.
[5]
(ii) Is it true that 11101000 \(\in A^*\)? Justify your answer.
[2]

(b) Consider the following binary relation ⊆, defined over the set of all
NFAs by setting, for any NFAs M1, M2:

M1 ⊆ M2 iff L(M1) ⊆ L(M2)

That is, M1 ⊆ M2 iff the language recognised by M1 is a subset of the
language recognised by M2. Is ⊆ an equivalence relation? Justify your
answer.
[6]
(c) Let A be the set of all strings over the alphabet 0, 1 that do not end
with more than two consecutive 0s. Examples of strings in A would
be: ϵ, 1, 100, 0110. Examples of strings not in A would be: 1000,
0011000000, 11101000.
(i) Is the following statement true or false?:
There is a regular expression representing A
You must fully justify your answer. Any definitions, theorems or
propositions from the lecture notes that you use must be clearly
stated in full.
[5]
(ii) Is it true that 11101000 ∈ A^*? Justify your answer.
[2]

Added by Sabrina A.

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