Given two sets $|A|=3,|P(B)|=64,|P(AB)|=256,$ what is the cardinality of each of these sets: $B,AB$ and $A-B,$ where and $x!$inB

Determine whether each of these statements is correct or incorrect. Justify your answers.

$($a$)$ xin$\{x\}$

$($b$)\{x\}sube\{x\}$

$($c$)\{x\}in\{x\}$

$($d$)\{x\}in\{\{x\}\}$

$($e$)sube\{x\}$

$($f$)in\{x\}$

Prove or disprove the following about sets:

$($a$)$ If AsubeB, then $AB=$A and $AB=B.$

$($b$)(A-B)(B-A)=AB.$

$($c$)A(BC)=(AB)(AC).$

$($d$)P(AB)P(A)P(B).$