The number of min-terms after minimizing the following Boolean expression is _______________ . $$$\left[ {D' + AB' + A'C + AC'D + A'C'D} \right]'$$$

A half adder is implemented with $$XOR$$ and $$AND$$ gates. A full adder is implemented with two half adders and one $$OR$$ gate. The propagation delay of an $$XOR$$ gate is twice that of an $$AND/OR$$ gate. The propagation delay of an $$AND/OR$$ gate is $$1.2$$ microseconds. A $$4$$-bit ripple-carry binary adder is implemented by using four full adders. The total propagation time of this $$4$$-bit binary adder in microseconds is____________ .

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

What is the Boolean expression for the output f of the combinational logic circuit of NOR gates given below?