Please provide a DFA M for this.

Consider a finite language $L$ with strings $\{"$a$","$b$","$ab$"\}.$

We construct a DFA M as follows:

States: $q_a,q_b,q_{ab},q_{\text{r}\text{e}\text{j}\text{e}\text{c}\text{t}}$

Transitions:

$q0--a-qa$

$q0--b-qb$

$q0--a-$qab

$q0--b-$ qreject

Accepting States: $q_a,q_b,q_{ab}$

Reject State: $q_{\text{r}\text{e}\text{j}\text{e}\text{c}\text{t}}$