Consider an eight-bit ripple-carry adder for computing the sum of $$A$$ and $$B,$$ where $$A$$ and $$B$$ are integers represented in $$2’s$$ complement form. If the decimal value of $$A$$ is one, the decimal value of $$B$$ that leads to the longest latency for the sum to stabilize is __________ .

Let, $${x_1} \oplus {x_2} \oplus {x_3} \oplus {x_4} = 0$$ where $${x_1},\,{x_2},\,{x_3},\,{x_4}$$ are Boolean Variables, and $$ \oplus $$ is the $$XOR$$ operator.

Which one of the following must always be TRUE?

Let $$X$$ be the number of distinct $$16$$-bit integers in $$2’s$$ complement representation. Let $$Y$$ be the number of distinct $$16$$-bit integers in sign magnitude representation.
Then $$X −Y$$ is ____________.

A binary relation $$R$$ on $$N \times N$$ is defined as follows: $$(a,b)R(c,d)$$ if $$a \le c$$ or $$b \le d.$$ Consider the following propositions:

$$P:$$ $$R$$ is reflexive
$$Q:$$ $$R$$ is transitive

Which one of the following statements is TRUE?