Closure properties

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

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

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

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