Implementation of AND Gate from NOR Gate
The NOR gate is functionally complete, which means every other gate can be built from it alone. This page shows how to build a AND gate using only NOR gates, why the construction works, and how many gates it costs.
Why AND can be built from NOR
The mirror image of building OR from NAND. De Morgan again: A AND B equals NOT((NOT A) OR (NOT B)).
Functional completeness is the reason chip designers care: a fab only has to get one gate right, and every circuit can be built from copies of it. NAND and NOR are both functionally complete; no other two-input gate is, on its own.
Step-by-step construction
Step 1
NOT A.
Step 2
Step 3
NOT((NOT A) OR (NOT B)), which equals A AND B.
The boolean expression
A AND B = NOR(NOT A, NOT B)
Truth table
| A | B | AND |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Build the circuit, then check every row. If all four match, the construction is correct: a truth table is a complete specification for a combinational circuit, so agreeing on every row means the two circuits are the same function.
Gate count
This construction uses 3 NOR gates. Gate count matters in real hardware because every gate costs area, power and a small propagation delay, which is why a circuit built entirely from one gate type is usually larger than the same logic built from mixed gates.
Build it yourself
Reading the construction is not the same as getting it working. Open the editor, drop in the gates, and watch the output change as you toggle the inputs.
Related pages
Every conversion follows the same two moves: get the un-inverted version, then cancel or add an inversion. Once you have seen AND from NAND and OR from NAND, the rest are variations. The wider digital logic track covers where these gates end up: adders, latches and memory.