GATE CS 1998 Q1.13 — Boolean Algebra

When a bit-string $B$ is XORed with itself $n$ times, we perform $n$ XOR operations, which involves $(n + 1)$ occurrences of $B$:

Using the fundamental properties of the XOR operation:

• $B \oplus B = 0$ (Self-inverse property)
• $B \oplus 0 = B$ (Identity property)

Let's evaluate the result for different values of $n$:

• For $n = 1$ (odd): $B \oplus B = 0$
• For $n = 2$ (even): $B \oplus B \oplus B = 0 \oplus B = B$
• For $n = 3$ (odd): $B \oplus B \oplus B \oplus B = 0 \oplus 0 = 0$
• For $n = 4$ (even): $B \oplus B \oplus B \oplus B \oplus B = 0 \oplus 0 \oplus B = B$

Thus, when $n$ is even, the expression simplifies to $B$, meaning the bit-string remains unchanged.