With Complements (1.3.1, 6)

Knuth asks

Prove or disprove the following rule for negating a n-bit number in two's complement notation: "Complement all the bits, then add one." (For example, ^#0...01 becomes ^#f...fe, then ^#f...ff; also ^#f...ff becomes ^#0..00, then ^#0..01.)

Solution.