GATE CS 2017 Set 1 Q9 — Number Representations (Fixed and Floating Point)

In $2$'s complement representation, an overflow occurs during addition if and only if the addition of two numbers with the same sign results in a sum with the opposite sign. Adding a positive number and a negative number can never result in an overflow.

For $8$-bit numbers, the sign bits (Most Significant Bits) are $A_7$, $B_7$, and the resulting sum sign bit is $S_7$.

There are two possible scenarios for overflow:

1. Adding two negative numbers results in a positive sum:
   • Both inputs are negative: $A_7 = 1$ and $B_7 = 1$
   • The sum is positive: $S_7 = 0$ (which means $\overline{S_7} = 1$)
   • This condition is represented by the boolean expression: $A_7 \cdot B_7 \cdot \overline{S_7}$
2. Adding two positive numbers results in a negative sum:
   • Both inputs are positive: $A_7 = 0$ and $B_7 = 0$ (which means $\overline{A_7} = 1$ and $\overline{B_7} = 1$)
   • The sum is negative: $S_7 = 1$
   • This condition is represented by the boolean expression: $\overline{A_7} \cdot \overline{B_7} \cdot S_7$

Combining these two mutually exclusive cases, the overall overflow condition is:

Thus, Option C is the correct expression.