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Peter Linz Edition 4 Exercise 4.1 Question 16 (Page No. 110)
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Show that if the statement “If $L_1$ is regular and $L_1 ∪ L_2$ is also regular, then $L_2$ must be regular“
were true for all $L_1$ and $L_2$, then all languages would be regular.
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Peter Linz Edition 4 Exercise 4.1 Question 26 (Page No. 110)
Let $G_1$ and $G_2$ be two regular grammars. Show how one can derive regular grammars for the languages (a) $L (G_1) ∪ L (G_2)$. (b) $L (G_1) L (G_2)$. (b) $L (G_1)^*$.
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Peter Linz Edition 4 Exercise 4.1 Question 22 (Page No. 110)
The $\textit{shuffle}$ of two languages $L_1$ and $L_2$ is defined as $\textit{shuffle}(L_1,L_2)= \{w_1v_1w_2v_2w_3v_3...w_mv_m:w_1w_2w_3….w_m∈L_1, v_1v_2...v_m∈L_2,\text{ for all }w_i,v_i∈Σ^*\}.$ Show that the family of regular languages is closed under the $shuffle$ operation.
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Peter Linz Edition 4 Exercise 4.1 Question 21 (Page No. 110)
Define $exchange(a_1a_2a_3...a_{n-1}a_n)=a_na_2a_3...a_{n-1}a_1$, and $exchange(L)=$ {$v:v=exchange(w)$ for some $w∈L$} Show that the family of regular languages is closed under exchange.
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Peter Linz Edition 4 Exercise 4.1 Question 20 (Page No. 110)
For a string $a_1a_2…a_n$ define the operation $shift$ as $shift(a_1a_2...a_n)=a_2a_3...a_na_1$ From this, we can define the operation on a language as $shift(L)=$ {$v:v=shift(w$ for some $w∈L$} Show that regularity is preserved under the shift operation.
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