For each of the following statements, state if they are true or false.

$($a$)$ A cache using a write$-$through policy writes back to the main memory simultaneously as it writes to the cache.

$($b$)$ Two important properties of a virtual memory are to give the illusion of a very large memory and to give memory protection between two concurrently running programs.

$($c$)$ A unit called the memory management unit $($MMU$)$ is a common solution to perform fast translations from virtual page numbers to physical page numbers.

$($d$)$ Modern high$-$performance processors do not use multi$-$level caches because they are too expensive and give little performance benefits.