Consider the 8-bit signed integers $X, Y$ and $Z$ represented using the sign-magnitude form. The binary representations of $X$ and $Y$ are as follows: $$X: 10110100 \quad Y: 01001100$$ Which of the following operations to compute $Z$ result(s) in an arithmetic overflow?

In an 8-bit sign-magnitude representation:

• The MSB is the sign bit ($0$ for positive, $1$ for negative).

• The remaining $7$ bits represent the magnitude.

• The range of representable numbers is from $-(2^7 - 1) = -127$ to $+(2^7 - 1) = +127$.

Given the binary representations:

• For $X = 10110100_2$:

  • Sign bit is $1$ (negative).

  • Magnitude bits are $0110100_2 = 32 + 16 + 4 = 52$.

  • Therefore, $X = -52$.

• For $Y = 01001100_2$:

  • Sign bit is $0$ (positive).

  • Magnitude bits are $1001100_2 = 64 + 8 + 4 = 76$.

  • Therefore, $Y = +76$.

Now, evaluate each operation for $Z$:

• Option (A): $Z = X + Y = -52 + 76 = +24$. Since $+24 \in [-127, +127]$, there is no overflow.

• Option (B): $Z = X - Y = -52 - 76 = -128$. Since $-128 < -127$, this value cannot be represented in 8-bit sign-magnitude format, resulting in an overflow.

• Option (C): $Z = -X + Y = -(-52) + 76 = 52 + 76 = +128$. Since $+128 > +127$, this value cannot be represented in 8-bit sign-magnitude format, resulting in an overflow.

• Option (D): $Z = -X - Y = 52 - 76 = -24$. Since $-24 \in [-127, +127]$, there is no overflow.

Thus, operations (B) and (C) result in arithmetic overflow.