$[36$ points, $4$ points per item$]$ Let A and B be two languages. Suppose A $<=$P B $($polynomial$-$time reduction$).$ Answer True or False for each of the following items:

$(1)$ If B is NP$-$complete and A is in NP$,$ then A is NP$-$complete. $(2)$ If A is NP$-$complete and B is in NP$,$ then B is NP$-$complete.

Let A and B be two languages. Suppose A $$ B$.$ Answer True or False for the following items: $(3)$ If B is NP$-$complete and A is in NP$,$ then A is NP$-$complete.

$(4)$ If A is NP$-$complete and B is in NP$,$ then B is NP$-$complete. Answer True or False for the following questions:

$(5)$ If a language L is decidable then it must be the case that both L and L are recognizable. $(6)$ If L is undecidable then it must be the case that exactly one of L or L is not recognizable. $(7)$ If both A and B are NP$-$complete, then we have both A $<=$P B and B $<=$P A$.$

$(8)$ TIME$($t$($n$))$ defined using single$-$tape DTMs is the same as using multi$-$tape DTMs$.$

$(9)$ The class P defined using single$-$tape DTMs is the same as using multi$-$tape DTMs$.$