Can “bitwise operations alone” reverse the sign of an integer directly (i.e., make x become -x) without any arithmetic step?

Introduction / Context:
This question distinguishes pure bit-twiddling from arithmetic semantics. In two’s-complement systems, the negative of x is defined as the two’s-complement of x, which is formed by inverting all bits and adding 1. The key point is that the final +1 is an arithmetic addition, not a bitwise operator.

Given Data / Assumptions:

• Two’s-complement integer representation (used by virtually all modern CPUs).
• Bitwise operators available: ~, &, |, ^, <<, >>.
• Goal: compute -x from x.

Concept / Approach:
Two’s complement is defined such that x + (~x + 1) = 0. The transformation “invert then add one” yields the negative. Using ~x alone produces the ones’-complement, which equals -x - 1, not -x. Similarly, simply toggling the sign bit or xoring with all ones does not account for carries and will fail for most values. Therefore, bitwise operations alone (without the addition step) cannot produce -x in general.

Step-by-Step Solution:

Verification / Alternative check:
Example: x = 5. ~x = …11111010 (for 8-bit illustration: 0xFA), which equals -6, not -5. Adding 1 gives 0xFB (…11111011), which equals -5. Example: x = 0 → ~0 = all ones (which is -1), and -0 is 0; you still need the +1 to get 0.

Why Other Options Are Wrong:

Common Pitfalls:
Confusing flipping the sign bit with full two’s-complement negation; forgetting the +1 step; using width-specific constants that harm portability.

Final Answer:
No — in two’s complement, sign reversal is ~x + 1, which requires an addition step.