Consider the following combinational function block involving four Boolean variables $$x, y, a,$$
$$b$$ where $$x, a, b$$ are inputs and $$y$$ is the output.
Which one of the following digital logic blocks is the most suitable for implementing this function?

Let $$ \oplus $$ denote the exclusive $$OR\left( {XOR} \right)$$ operation. Let $$'1'$$ and $$'0'$$ denote the binary constants. Consider the following Boolean expression for $$F$$ over two variables $$P$$ and $$Q$$:
$$F\left( {P,Q} \right) = \left( {1 \oplus P} \right) \oplus \left( {P \oplus Q} \right) \oplus \left( {P \oplus Q} \right) \oplus \left( {Q \oplus 0} \right)$$

The equivalent expression for $$F$$ is

Consider the following interm expression of $$F:$$
$$F\left( {P,\,Q,\,R,\,S} \right) = \sum {0,2,5,7,8,10,13,15} $$
The minterms $$2, 7, 8$$ and $$13$$ are 'do not care' terms. The minimal sum-of-products form for $$F$$ is _______

The above synchronous sequential circuit built using $$JK$$ flip-flops is initialized with $${Q_2}{Q_1}{Q_0} = 000.\,\,$$ The state sequence for this circuit for the next $$3$$ clock cycles is