Consider the two lists List-I and List-II given below:
| List - I | List - II | ||
|---|---|---|---|
| (i) | Context free languages | (a) | Closed under union |
| (ii) | Recursive languages | (b) | Not closed under complementation |
| (iii) | Regular languages | (c) | Closed under intersection |
For matching of items in List-I with those in List-II, which of the following option(s) is/ are CORRECT?
Let $\Sigma=\{a, b, c\}$. For $x \in \Sigma^{\star}$, and $\alpha \in \Sigma$, let $\#_\alpha(x)$ denote the number of occurrences of a in $x$. Which one or more of the following option(s) define(s) regular language(s)?
Let $\Sigma=\{1,2,3,4\}$ For $x \in \Sigma^*$, let prod $(x)$ be the product of symbols in $x$ modulo 7 . We take $\operatorname{prod}(\varepsilon)=1$, where $\varepsilon$ is the null string.
For example, $\operatorname{prod}(124)=(1 \times 2 \times 4) \bmod 7=1$.
Define $L=\left\{x \in \Sigma^{\star} \mid \operatorname{prod}(x)=2\right\}$.
The number of states in a minimum state DFA for $L$ is _________ (Answer in integer)
Despite his initial hesitation, Rehman's __________ to contribute to the success of the project never wavered.
Select the most appropriate option to complete the above sentence.