Give a short argument that the complexity class $P$ is closed under Cartesian

product, that is$,$ if A and $B$ are languages in $P,$ then $AB$ would be in $P$ also.

We supply again the definition of $P$ here:

$P=u_k$TIME$(n^k).$

$[$HINT: If the language A and $B$ were in $P,$ both their decider TMs would

function in polynomial time. A decider for $AB$ could be constructed from the

deciders for A and $B$ by running each one to decide if the components of an

ordered pair $(a,b)$ in $AB$ are, in fact, in A and $B,$ respectively.$](10$ pts$.)$