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

For a $$3$$ -input logic circuit shown below, the output $$Z$$ can be expressed as

The Boolean expression $$AB + A\overline C + BC$$ simplifies to

$$f\left( {A,\,B,\,C,\,D} \right) = \Pi M\left( {0,1,3,4,5,7,9,11,12,13,14,15} \right)$$ is a Maxterm representation of a Boolean function $$f(A,B,C,D)$$ where $$A$$ is the $$MSB$$ and $$D$$ is the $$LSB$$. The equivalent minimized representation of this function is

Consider the following Sum of products expression, $$F.$$
$$F = ABC + \overline A \overline B C + A\overline B C + \overline A BC + \overline A \overline B \overline C $$

The equivalent Product of Sums expression is

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

A Boolean function is given as

$$ f=(\bar{u}+\bar{v}+\bar{w}+\bar{x}) \cdot(\bar{u}+\bar{v}+\bar{w}+x) \cdot(\bar{u}+v+\bar{w}+\bar{x}) \cdot(\bar{u}+v+\bar{w}+x) $$

The simplified form of this function is represented by

The logical gate implemented using the circuit shown below where $${V_1}$$ and $${V_2}$$ are inputs (with $$0$$ $$V$$ as digital $$0$$ and $$5$$ $$V$$ as digital $$1$$) and $${V_{OUT}}$$ is the output is

The simplified form of the Boolean expression $$Y = \left( {\overline A BC + D} \right)\left( {\overline A D + \overline B \overline C } \right)$$ can be written as

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