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Design a machine of the following language: L = Containing even a where ∑ = {a, b}

Answer: 

Here L = {λ, b, aa, aabb, aaaabbbb, ……}

As per the question,

Here r indicates remainder when the number of ‘a’ is even remainder is zero i.e. r = 0 otherwise remainder is one i.e. r = 1 (when the number of ‘a’ is odd).

At ${\mathit{q}}_{\mathbf{0}}$, the number of ‘a’ is always even that’s why we assign r = 0 at ${\mathit{q}}_{\mathbf{0}}$ state.

But at ${\mathit{q}}_{\mathbf{1}}$, the number of ‘a’ is always odd that’s why we assign r = 1 a ${\mathit{q}}_{\mathbf{1}}$ state.

${\mathit{q}}_{\mathbf{0}}$ is the final state because the machine is for even ‘a’. 

Same as above machine we can design even number of ‘b’:

Regular Expression for even number of b’s $\mathit{R}{\mathit{E}}_{\mathbf{1}}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{(}\mathit{a}\mathbf{*}\mathit{b}\mathit{a}\mathbf{*}\mathit{b}\mathit{a}\mathbf{*}\mathbf{)}\mathbf{*}$