Let $$ \ne $$ be a binary operator defined as $$X \ne Y = X' + Y'$$ where $$𝑋$$ and $$𝑌$$ are Boolean variables. Consider the following two statements. $$$\eqalign{ & \left( {S1} \right)\,\,\,\,\,\,\,\left( {P \ne Q} \right) \ne R = P \ne \left( {Q \ne R} \right) \cr & \left( {S2} \right)\,\,\,\,\,\,\,Q \ne R = R \ne Q \cr} $$$

Which of the following is/are true for the Boolean variables $$𝑃, 𝑄$$ and $$𝑅$$?

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
$$f\left( {w,x,y,z} \right) = \sum {\left( {0,2,4,5,6,10} \right)} $$ _________________.