Suppose $M_1$ and $M_2$ are two Turing machines such that $L(M_1)\subseteq L(M_2)$. Which of the following is true?
- (A) On every input on which $M_1$ does not halt, M2 does not halt
- (B) On every input on which $M_1$ halts, $M_2$ halts too
- (C) On every input which $M_1$ accepts, $M_2$ halts.
- (D)On every input which $M_2$ accepts, $M_1$ halts.
I am confused between B and C. This was an online practice test question.
For (B) my claim is that when $M_1$ is able to decide the language (i.e., $M_1$ halts on every input), then, since it is given $L(M_1) \subseteq L(M_2)$, $M_2$ should also halt and be able to decide on every input.
However, option (C) also looks convincing. If for every input $M_1$ says "yes" in the language, then $M_2$ should also be able to decide for that input.
Please let me know how to approach this correctly.
EDIT: After reading the discussion, am I correctly interpreting the problem If I imagine the figure below
SO, (B) must be the wrong choice according to this right?