Consider a carry lookahead adder for adding two $$n$$-bit integers, built using gates of fan-in at most two. The time to perform addition using this adder is

Consider the two cascaded $$2$$-to-$$1$$ multiplexers as shown in the figure.

The minimal sum of products form of the output $$X$$ is

We want to design a synchronous counter that counts the sequence $$0-1-0-2-0-3$$ and then repeats. The minimum number of $$J-K$$ flip-flops required to implement this counter is _________.

Consider the Boolean operator $$ \ne $$ with the following properties:
$$x \ne 0 = x,\,\,x \ne 1 = \overline x ,\,\,x \ne x = 0$$ and $$x \ne \overline x = 1.$$ Then $$x \ne y$$ is equivalent to