A 4-bit parallel adder uses four full adders. What is the maximum number of bits in the sum output, including the carry-out bit?

Explanation:

4-Bit Parallel Adder and Its Output:

A 4-bit parallel adder is a digital circuit used for adding binary numbers. It consists of four full adders connected in series, where each full adder is responsible for adding corresponding bits from two binary numbers along with a carry from the previous stage. This configuration allows for the addition of two 4-bit binary numbers.

Understanding the Sum Output:

In a parallel adder, the number of bits in the sum output depends on two factors:

• The number of bits in the numbers being added.
• The carry-out bit from the most significant bit (MSB).

When two 4-bit binary numbers are added, the sum can theoretically require 5 bits to represent the result, because:

• A 4-bit binary number has a maximum value of 1111 (in binary), which equals 15 in decimal.
• The sum of two maximum 4-bit binary numbers (15 + 15) equals 30 in decimal, which is represented as 11110 in binary—a 5-bit result.
• The fifth bit arises from the carry-out of the MSB addition.

Detailed Analysis:

The operation of the 4-bit parallel adder can be broken down as follows:

• Input: Two 4-bit binary numbers (A and B) and an initial carry-in (C_in, typically 0).
• Output: A 4-bit sum (S₃, S₂, S₁, S₀) and a carry-out (C_out) from the MSB addition.

The sum bits (S₀, S₁, S₂, S₃) represent the 4-bit result of the addition, while the carry-out (C_out) represents the fifth bit, indicating an overflow from the MSB. Thus, the total number of bits in the sum output, including the carry-out, is 5.

Why Option 1 is Correct:

Option 1 states that the maximum number of bits in the sum output, including the carry-out bit, is five. This is correct because:

• The addition of two 4-bit numbers can produce a result requiring 5 bits to represent.
• The fifth bit arises from the carry-out generated during the addition of the MSB.
• Therefore, the sum output consists of 4 bits from the addition itself and an additional carry-out bit, making a total of 5 bits.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: Three bits.

This option is incorrect because a 4-bit parallel adder must produce at least a 4-bit result for the sum of two 4-bit binary numbers. Including the carry-out bit, the total number of bits in the sum output is 5, not 3.

Option 3: Eight bits.

This option is incorrect because a 4-bit parallel adder is designed for adding two 4-bit numbers, not two 8-bit numbers. The addition of two 4-bit numbers results in a sum output requiring a maximum of 5 bits (4 bits for the sum and 1 bit for the carry-out). An 8-bit sum output would require an entirely different circuit, such as an 8-bit adder.

Option 4: Four bits.

This option is partially correct but does not account for the carry-out bit. While the sum of two 4-bit numbers produces a 4-bit result, the carry-out from the MSB adds an additional bit, making the total number of bits in the sum output 5.

Conclusion:

In a 4-bit parallel adder, the maximum number of bits in the sum output, including the carry-out bit, is five. This result accounts for the 4-bit sum and the additional carry-out bit from the MSB addition. Understanding the operation of parallel adders is essential for analyzing binary addition circuits and their outputs.