$L_1$ = $\{\langle M \rangle \mid M$ is a turing machine and $M$ halts on some string$\}$
$L_2$ = $\{\langle M \rangle \mid M$ is a turing machine and $M$ halts on all strings $\}$
a) Is $L_2$ a superset of $L_1$ ?
b) Is $L_2$ not co-recognizable and not recognizable ?
(without formally proving)
my attempt
$a)$ $L_2$ is a superset of $L_1$ since all strings are in $L_2$ whereas $L_1$ is some string
$(b)$ $L_2$ is not recognizable because, out of the infinite enumerations of strings, a single one could not halt disproving it from being recognizable.
For not co-recognizable: It can be written like this
$\bar{L_2} = \{ \langle M \rangle \mid M$ is a turing machine and $M$ loops on some string $\}$.
It's not possible to make a recognizer for loops so its instantly not co-recognizable. This is because looping means it never stops so to come up with a recognizer for anything to prove it loops is impossible since we are working with infinite combinations.
Not sure about it being a superset or not and my explanation for co-recognizable