$(20$ points$)$ Circle True or False. Justify your answers. If false, state a counterexample, If true

prove your answer. $($Answers without justification won't be graded$).$

a$)$ If $L$ is regular, then $L^{\text{'}}=\{aw|winL\}$ is regular.

True False

b$)$ For every regular language $L,$ every subset of $L$ is regular as well.

True False

c$)$ The intersection of any two non$-$regular languages is non$-$regular.

True False

d$)$ Every subset of a regular language is regular.

True False