Draw a DFA $($Deterministic Finite Automaton$)$ that recognizes exactly the regular language

$$

L defined as

$$

$=$

$\{$

$$

$$

$010$

or

$101$

is a substring of

$$

$\}$

L$=\{$w$010$ or $101$ is a substring of w$\}.$

b$)$ Prove by mathematical induction that the DFA provided in part $($a$)$ recognizes exactly the language

$$

L$.$