Consider the languages $${L_1} = \phi $$ and $${L_2} = \left\{ a \right\}.$$ Which one of the following represents $${L_1}\,L_2^ * UL_1^ * ?$$

Consider the following languages
$${L_1} = \left\{ {{0^p}{1^q}{0^r}\left| {p,q,r \ge 0} \right.} \right\}$$
$${L_2} = \left\{ {{0^p}{1^q}{0^r}\left| {p,q,r \ge 0,p \ne r} \right.} \right\}$$

Which one of the following statements is FALSE?

Consider the $$DFA$$ $$A$$ given below.

Which of the following are FALSE?
$$1.$$ Complement of $$L(A)$$ is context - free.
$$2.$$ $$L(A)$$ $$ = \left( {{{11}^ * }0 + 0} \right)\left( {0 + 1} \right){}^ * {0^ * }\left. {{1^ * }} \right)$$
$$3.$$ For the language accepted by $$A, A$$ is the minimal $$DFA.$$
$$4.$$ $$A$$ accepts all strings over $$\left\{ {0,1} \right\}$$ of length at least $$2.$$

Which of the following statements is/are FALSE?
$$1.$$ For every non-deterministic Turing machine, there exists an equivalent deterministic Turing machine
$$2.$$ Turing recognizable languages are closed under union and complementation
$$3.$$ Turing decidable languages are closed under intersection and complementation
$$4.$$ Turing recognizable languages are closed under union and intersection