# Superset of another language and recognizability of turing machines

$$L_1$$ = $$\{\langle M \rangle \mid M$$ 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