Closure properties

Exercise 1. Show that $(10 \cup 0)^*(1 \cup \epsilon)$ and $(1 \cup \epsilon)(00^*1)^*0^*$ are equivalent regular expressions.

Exercise 2. Write the “complement” regular expression for $(10 \cup 0)^*(1 \cup \epsilon)$.

Exercise 3. Define the reversal of a language $ L $ as $ rev(L) = \{ w \in \Sigma^* | rev(w) \in L \}$, where $ rev(c_1c_2 \dots c_n) = c_nc_{n - 1} \dots c_1$ , with $ c_i \in \Sigma, 1 \leq i \leq n $. Show that reversal is a closure property.

Exercise 4. Let $ L \subseteq \Sigma^* $ be a language and $ c \in \Sigma $ a symbol. The quotient of $ L $ and $ c $ is the language defined as $ L/c = \{ w \in \Sigma^* | wc \in L\} $.

4.1. Let $ L = (aaa \cup ba)^*(ab)^* $. What is the language $ L/a $ ?

4.2. Prove that if $ L $ is a regular language, then $ L/c$ is a regular language $ \forall c \in \Sigma$.

4.3. Let $ L \subseteq \Sigma^* $ be a language and $ c \in \Sigma $ a symbol. Then $ c / L = \{ w \in \Sigma^* | cw \in L \} $. Prove that if $ L $ is a regular language, then $ a / L $ is a regular language,$ \forall a \in \Sigma$.

Exercise 5. Prove that $ \text{suffix}(L) = \{ w \in \Sigma^* | \exists x \in \Sigma^*, \: \text{such that} \: xw \in L \} $ is a closure property.

Exercise 6. Define $ min(L) = \{ w \in L | \; \nexists x \in L, \; y \in \Sigma^* \setminus \{\epsilon\}, \: \text{such that} \: xy = w \}$.

Example: If $ L = \{ aab, bab, aa \} $, then $ min(L) = \{ bab, aa \} $.

6.1. What is the language $ min(L(a^*)) $ ?

6.2. What is the language $ min(L(a^*b)) $ ?

6.3. Prove that $ min $ has the closure property with respect to the regular languages.