Consider the relation $$X\left( {P,Q,R,S,T,U} \right)$$ with the following set of functional dependencies
$$\eqalign{ & F = \left\{ \, \right. \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {P,R} \right\} \to \left\{ {S,T} \right\}, \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {P,S,U} \right\} \to \left\{ {Q,R} \right\} \cr & \,\,\,\,\,\,\,\,\,\,\left. \, \right\} \cr} $$
Which of the following is the trivial functional dependency in $${F^ + },$$ where $${F^ + }$$ is closure of $$f$$ ?

Given the function $$F = P′ + QR,$$ where $$F$$ is a function in three Boolean variables $$P,Q$$ and $$R$$ and $$P'=!P,$$ consider the following statements.

$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {S1} \right)\,\,\,\,F = \sum {\left( {4,5,6} \right)} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {S2} \right)\,\,\,\,F = \sum {\left( {0,1,2,3,7} \right)} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {S3} \right)\,\,\,\,F = \sum {\Pi \left( {4,5,6} \right)} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {S4} \right)\,\,\,\,F = \sum {\Pi \left( {0,1,2,3,7} \right)} \cr} $$

Which of the following is true?

Consider the equation $${\left( {43} \right)_x} = {\left( {y3} \right)_8}$$ where $$x$$ and $$y$$ are unknown. The number of possible solutions is ______________

The total number of prime implicants of the function
$$f\left( {w,x,y,z} \right) = \sum {\left( {0,2,4,5,6,10} \right)} $$ _________________.