A circuit outputs a digit in the form of $$4$$ bits. $$0$$ is represented by $$0000$$, $$1$$ by $$0001..., $$ $$9$$ by $$1001.$$ A combinational circuit is to be designed which takes these $$4$$ bits as input and outputs $$1$$ if the digit $$ \ge 5,$$ and $$0$$ otherwise. If only $$AND,$$ $$OR$$ and $$NOT$$ gates may be used, what is the minimum number of gates required?

Consider a multiplexer with $$X$$ and $$Y$$ as data inputs and $$Z$$ as control input. If $$z=0$$ select input $$x,$$ and $$z=1$$ selects input $$Y$$. what are the connections required to realize the $$2-$$variable Boolean function $$f=T+R$$ without using any additional Hardware?

$$SR.$$ latch made by cross coupling two $$NAND$$ gates if $$S=R=0,$$ Then it will result in

Consider the partial implementation of a $$2$$-bit counter using $$T$$ flip-flops following the sequence $$0$$-$$2$$-$$3$$-$$1$$-$$0,$$ as shown below.

To complete the circuit, the input $$X$$ should be