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Design a machine of the following language: 

Answer:

As per the language given is,

So, the number of states will require to design automata, as we know it may 2 * 4 = 8 (mod 2 and mod 4) grade of states. 

But actually, it will not require 2 * 4 = 8 grade of states, it is some of the exceptional case where the grade is not calculated as mod n x mod m. 

If there is a common factor between n and m (mod n and mod m) then the grade of states will not be calculated as n x m.

$\mathbf{\left(}{\mathbf{n}}_{\mathbf{a}}\mathbf{\right(}\mathbf{\omega }\mathbf{)}\mathbf{ }\mathbf{mod}\mathbf{ }\mathbf{2}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{)}$ = (0, 2, 4, 6, 8, 10, 12…….)  AND  $\mathbf{\left(}{\mathbf{n}}_{\mathbf{a}}\mathbf{\right(}\mathbf{\omega }\mathbf{)}\mathbf{ }\mathbf{mod}\mathbf{ }\mathbf{4}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{)}$ = (0, 4, 8, 12………)

So, some of the element is common for both cases and there is AND  so it is intersection.

So, if we do intersection (0, 2, 4, 6, 8, 10, 12…….) ∩ (0, 4, 8, 12………) = (0, 4, 8, 12…..)

                                                                        Or

So, it only requires designing automata of $\mathbf{\left(}{\mathbf{n}}_{\mathbf{a}}\mathbf{\right(}\mathbf{\omega }\mathbf{)}\mathbf{ }\mathbf{mod}\mathbf{ }\mathbf{4}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{)}$ for language,

But the machine will be changed if the language has slightly changed as,

Now,

$\mathbf{\left(}{\mathbf{n}}_{\mathbf{a}}\mathbf{\right(}\mathbf{\omega }\mathbf{)}\mathbf{ }\mathbf{mod}\mathbf{ }\mathbf{2}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{)}$ = (0, 2, 4, 6, 8, 10, 12….)  OR $\mathbf{\left(}{\mathbf{n}}_{\mathbf{a}}\mathbf{\right(}\mathbf{\omega }\mathbf{)}\mathbf{ }\mathbf{mod}\mathbf{ }\mathbf{4}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{)}$ = (0, 4, 8, 12………)

So, some of the element is common for both cases and there is OR so it is Union. And another thing that is clear from above is $\mathbf{\left(}{\mathbf{n}}_{\mathbf{a}}\mathbf{\right(}\mathbf{\omega }\mathbf{)}\mathbf{ }\mathbf{mod}\mathbf{ }\mathbf{2}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{)}$ ⊇ $\mathbf{\left(}{\mathbf{n}}_{\mathbf{a}}\mathbf{\right(}\mathbf{\omega }\mathbf{)}\mathbf{ }\mathbf{mod}\mathbf{ }\mathbf{4}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{)}$ i.e. mod 2 is super-set of mod 4.

So if we do union (0, 2, 4, 6, 8, 10, 12….) U (0, 4, 8, 12………) = (0, 2, 4, 6, 8, 10, 12….)

                                                                       Or

So, it only requires designing automata of $\mathbf{\left(}{\mathbf{n}}_{\mathbf{a}}\mathbf{\right(}\mathbf{\omega }\mathbf{)}\mathbf{ }\mathbf{mod}\mathbf{ }\mathbf{2}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{)}$ for language,