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

Which of the following is true for the language $$\left\{ {{a^p}} \right.\left| P \right.$$ prime $$\left. \, \right\}$$?

If the strings of a language $$L$$ can be effectively enumerated in lexicographic (i.e., alphabetic$$(c)$$ order, which of the following statements is true?

Nobody knows yet if $$P=NP$$. Consider the language $$L$$ defined as follows
$$L = \left\{ {\matrix{ {{{\left( {0 + 1} \right)}^ * }\,\,\,if\,\,P = NP} \cr {\,\,\,\,\,\,\,\phi \,\,\,\,Otherwise} \cr } } \right.$$

Which of the following statement is true?