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2's complement subtraction; AND using NOR

Short answer

Subtraction: 2's complement of 111 is 001; 110 + 001 = 111, no carry, negative, magnitude 001 → −1. AND using NOR: two NOR gates as NOT invert A,B; a third NOR gates their outputs to give (A'+B')' = A·B.

Subtract (111)2 from (110)2 using 2's complement method. Draw the circuit of AND gate using NOR gates only.

Subtraction: (110)₂ − (111)₂ using 2's complement

The question asks to subtract 111 from 110 (6 − 7), using 3 bits.

  1. 1's complement of 111 = 000; 2's complement = 000 + 1 = 001.
  2. Add: 110 + 001 = 111.
  3. No end carry, so the result is negative.
  4. Magnitude = 2's complement of 111 = 000 + 1 = 001.

Answer: 110 − 111 = −(001)₂ = −1

AND gate using only NOR gates

Using De Morgan's theorem: A·B = (A' + B')' = NOR(A', B').

  • A NOR gate with both inputs joined acts as a NOT gate.
  • So two NOR gates (as NOT) invert A and B, and a third NOR gate combines them.
 A ---[NOR]--- A' ---+
      (inputs tied)  [NOR]---- Y = (A' + B')' = AB
 B ---[NOR]--- B' ---+
      (inputs tied)

Three 2-input NOR gates are needed.

✓ Answer checked by Claude Opus 5.5 (high effort)

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