## Design a machine of the following language:

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 statesit 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.

= (0, 2, 4, 6, 8, 10, 12…….)  AND   = (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  for language,

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

Now,

= (0, 2, 4, 6, 8, 10, 12….)  OR  = (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  ⊇  i.e. mod 2 is super-set of mod 4.

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

Or

So, it only requires designing automata of  for language,