$(40$ pts$)$ Let $x,$Ysube${}^{+}$be codes. For every true statement below present a proof. Otherwise, present a

counterexample. Determine whether:

$($a$)xY$ is a code

$($b$)xY$ is a code

$($c$)xY$ is a code

$($d$)Z$ is a code for Zsubex.