Given that language $${L_1}$$ is regular and that the language $${L_1} \cap {L_2}$$ is regular is the language $${L_2}$$ is always regular?

Is the class of regular sets closed under infinite union? Explain.

Give minimal $$DFA$$ that performs as a Mod-$$3$$ $$1's$$ counter, i.e., outputs a $$1$$ each time the number of $$1's$$ in the input sequence is a sequence is a multiple of $$3.$$

Give the regular expression over $${\left\{ {0,\,\,1} \right\}}$$ to denote the set of proper non-null substrings of the string $$0110$$.