Distinguishing One String from Another $($cont$\text{'}$d$.)$

In FA that accepts $\{b),$ we notice first that null string $$ is not in $L,b$ is in $L,$ string $a$ is not, and A and $u$ are $L-$distinguishable because Mab e $L$ and aabinL. So$,$ for that we need the following three states: