Let A and $B$ be languages. The difference of A and $B$ is the language

$A-B=\{xin{}^{**}|xinA$ and $x!$inB$\}.$

The symmetric difference of A and $B$ is the language

$AB=\{xin{}^{**}|xisin$ exactly one $ofAorB\}=(A-B)(B-A).$

Prove the following statements:

$($a$)$ If A and $B$ are regular, then $A-B$ is also regular.

$($b$)$ If A and $B$ are regular, then $AB$ is also regular.

Given DFAs for A and $B,$ explain how to construct DFAs for $A-B$ and $AB.$