Prove that $A/B$ is regular when $A$ is regular and $B$ is regular or not regular.

Question

Prove that $A/B$ is regular when $A$ is regular and $B$ is regular or not regular. $$A/B=\{w: wx\in A\ \text { for some }\ x\in B\}$$

Please give me a clue.

Answer 1

Hint. Observe that $$ A/B = \bigcup_{x \in B} A/x $$ and now use the fact that $A$ is regular.