Understanding Signed-Complement Arithmetic Addition

This question requires performing arithmetic addition of decimal numbers using the signed-complement system, primarily the 2's complement for negative numbers. We then need to match the resulting binary representations with those provided in List-II.

Case (A): Addition of $+6$ and $+13$

First, we convert the decimal numbers $+6$ and $+13$ into their 8-bit binary representations:

• The decimal number $+6$ is represented in 8-bit binary as $00000110$.
• The decimal number $+13$ is represented in 8-bit binary as $00001101$.

Now, we perform the addition:

The result of the addition is $00010011$. This binary number represents the decimal value $+19$. This matches representation (II) in List-II.

Case (B): Addition of $-6$ and $+13$

We convert the decimal numbers $-6$ and $+13$ into their 8-bit binary representations. For $-6$, we use the 2's complement:

• The decimal number $+13$ is represented in 8-bit binary as $00001101$.
• To find the 2's complement representation for $-6$:
  • The binary for $+6$ is $00000110$.
  • Invert the bits (1's complement): $11111001$.
  • Add 1 to the 1's complement: $11111001$ + 1 = $11111010$. This is the 2's complement for $-6$.

Now, we perform the addition:

Discarding the carry-out bit (the 9th bit), the result is $00000111$. This binary number represents the decimal value $+7$. This matches representation (I) in List-II.

Case (C): Addition of $+6$ and $-13$

We convert the decimal numbers $+6$ and $-13$ into their 8-bit binary representations. For $-13$, we use the 2's complement:

• The decimal number $+6$ is represented in 8-bit binary as $00000110$.
• To find the 2's complement representation for $-13$:
  • The binary for $+13$ is $00001101$.
  • Invert the bits (1's complement): $11110010$.
  • Add 1 to the 1's complement: $11110010$ + 1 = $11110011$. This is the 2's complement for $-13$.

The arithmetic addition of $+6$ and $-13$ results in a binary value that matches representation (IV) in List-II.

Let's examine representation (IV): $11111001$.

To convert this 8-bit 2's complement binary number back to its decimal value:

• Invert the bits: $11111001$ becomes $00000110$.
• Add 1 to the inverted bits: $00000110$ + 1 = $00000111$.
• The value $00000111$ represents the decimal number $7$. Therefore, the original binary number $11111001$ represents $-7$.

Thus, for case (C), the addition of $+6$ and $-13$ is matched with representation (IV).

Case (D): Addition of $-6$ and $-13$

We convert the decimal numbers $-6$ and $-13$ into their 8-bit binary representations using the 2's complement system:

• The 2's complement representation for $-6$ is $11111010$.
• The 2's complement representation for $-13$ is $11110011$.

Now, we perform the addition:

Discarding the carry-out bit, the result is $11101101$. This binary number represents the decimal value $-19$. This matches representation (III) in List-II.

Final Matching Summary

Based on the calculations and matching with List-II:

• (A) $+6, +13$ corresponds to (II) $00010011$ ($ +19 $).
• (B) $-6, +13$ corresponds to (I) $00000111$ ($ +7 $).
• (C) $+6, -13$ corresponds to (IV) $11111001$ ($ -7 $).
• (D) $-6, -13$ corresponds to (III) $11101101$ ($ -19 $).

The correct combination is (A) - (II), (B) - (I), (C) - (IV), (D) - (III). This corresponds to Option 2.