Bit manipulation

Begin by clarifying whether integers are signed, fixed-width, and whether negative values are in scope, because right shift behavior and sign bits vary by language. Bitwise AND tests whether a mask's bits are present, OR sets bits, XOR cancels equal bits and is useful when every other value appears exactly twice, and x plus one can be formed with carry logic but has overflow limits. Use parentheses around shifts and masks, and avoid shifting by an amount outside the type width. For counting set bits, repeatedly clear the lowest set bit with x = x AND (x minus one), performing one iteration per set bit. For a subset mask, state the invariant that bit i corresponds to item i. Test zero, powers of two, all bits set, and the highest bit. A fixed-width bit operation is O(1); Kernighan counting is O(k) time for k set bits and O(1) extra space.