Task $1$: Design a Regular Expression over for the following languages:

$D_1$ accepts symbols $($days$)$ from $1$ to $28($no leading zero$)$

$D_2$ accepts symbols $($days$)$ from $1$ to $30($no leading zero$)$

$D_3$ accepts symbols $($days$)$ from $1$ to $31($no leading zero$)$

$M_1$ accepts months that have only $28$ days $(2)$ excluding leap years

$M_2$ accepts months that have only $30$ days $(4,6,9,11)$

$M_3$ accepts months that have $31$ days $(1,3,5,7,8,10,12)$

Task $2$: Using Thompson's construction convert this regular expression $($by constructing the previous languages$)$ to a Non$-$deterministic Finite

Automata:

Date $=(D_1\frac{\text{@}}{\text{@}}{M}_1)+(D_2\frac{\text{@}}{\text{@}}{M}_2)+(D_3\frac{\text{@}}{\text{@}}{M}_3)$

Task $3$: Using Subset construction convert the previous Non$-$deterministic Finite Automata to a Deterministic Finite Automata.

Task $4$: Using Hopcroft's Algorithm minimize the previous Deterministic Finite Automata if possible.