For a three-bit word A, B, C, the parity bit P is defined as P = A ⊕ B ⊕ C; P is 1 if the number of ones is odd.

For a three-bit word A, B, C, the parity bit P is defined as P = A ⊕ B ⊕ C; P is 1 if the number of ones is odd.

• A. P = A AND B AND C
• B. P = A ⊕ B ⊕ C; P is 1 if the number of ones is odd
• C. P = A OR B OR C
• D. P = NOT(A ⊕ B ⊕ C)

Answer: (B)

To detect parity for three bits, you want the output to be 1 whenever there are an odd number of 1s among A, B, and C. The exclusive OR operation does exactly that: it outputs 1 when an odd number of inputs are 1. For three inputs, A ⊕ B ⊕ C yields 0 when there are 0 or 2 ones, and yields 1 when there are 1 or 3 ones. This matches the requirement that P is 1 if the number of ones is odd. The result is the same whether you evaluate as (A ⊕ B) ⊕ C or A ⊕ (B ⊕ C) because XOR is associative. The other forms don’t fit: AND would require all three to be 1 to get a 1; OR would be 1 as soon as any single input is 1, not depending on the count; NOT of XOR would give 1 when the number of 1s is even, not odd. So P = A ⊕ B ⊕ C is the correct expression.

To detect parity for three bits, you want the output to be 1 whenever there are an odd number of 1s among A, B, and C. The exclusive OR operation does exactly that: it outputs 1 when an odd number of inputs are 1. For three inputs, A ⊕ B ⊕ C yields 0 when there are 0 or 2 ones, and yields 1 when there are 1 or 3 ones. This matches the requirement that P is 1 if the number of ones is odd. The result is the same whether you evaluate as (A ⊕ B) ⊕ C or A ⊕ (B ⊕ C) because XOR is associative. The other forms don’t fit: AND would require all three to be 1 to get a 1; OR would be 1 as soon as any single input is 1, not depending on the count; NOT of XOR would give 1 when the number of 1s is even, not odd. So P = A ⊕ B ⊕ C is the correct expression.