An index set is a set of all indices of some family of computably enumerable sets.  It is known that the empty set is an index set and that $K = \{e \mid e \in W_e\}$ is not an index set.The following was an exam question I didn't manage to answer:Which of the following sets are index sets?1)  $\emptyset $2)  $K= \{ x \mid x \in W_x \}$ 3) $E = \{2,4,6,8, \dots \}$ 4) $\{ x \mid \phi_x = \phi_{x^2}\}$I think:It is known $K$ (i.e., the halting set) is not an index set so(2) is ruled out.  I read on this on last page problem 4 that the setof even numbers is an index set, so I'm not sure that (1) and (3) istrue!  But the answer sheet wrote just (1) is true.  I don't know whatis the definition of $E$ in (3) and not sure about (4). Can anyone clarify me in order to better understand this challenging question?  Why are (3) and (4) not index sets?  I'm not familiar with notation (4), any description? 
Are these sets of indices also index sets?
computability