$($INDIVIDUAL: Loopy TM$)$ Consider the following computational problem: Given the descrip$-$

tion of a Turing machine $M,$ is it the case that $M$ loops forever on every input $w?$

$($a$)$ Formulate this problem as language $\frac{L}{O}O{P}_{TM}.$

$($b$)$ Prove that $\frac{L}{O}O{P}_{TM}$ is undecidable. You may do this by either providing an "informal reduc$-$

tion" as on Homework $7,$ or by giving a mapping reduction $-$ it's your choice. In either case,

make sure to explain why the TM computing your reduction is correct.