logic minimization

Simplified form of the Boolean function $$ F(P, Q, R, S)=\bar{P} \bar{Q}+\bar{P} Q S+P \bar{Q} \bar{R} \bar{S}+P \bar{Q} R \bar{S} $$ is

In the sum of products function $$f\,\left( {X,\,Y,\,Z} \right) = \sum \left( {2,\,\,3,\,\,4,\,\,5} \right),$$ the prime implicants are

The Boolean expression $$X\overline Y Z + XYZ + \overline X Y\overline Z + \overline X \overline Y Z + XY\overline Z $$ can be simplified to