One-bit comparator

Is A greater, equal, or less than B? Exactly one lamp will tell you. A ready-made arithmetic circuit you can open in the TorchAnvil simulator.

Three questions, one circuit

Given two bits, there are only three possible relationships between them: A is bigger, A equals B, or A is smaller. A one-bit comparator answers all three at once, lighting exactly one of three lamps for any input.

How it works

Each lamp has its own tiny formula:

• A > B = A AND NOT B. The only way A beats B in one bit is A = 1 and B = 0.
• A = B = A XNOR B. XNOR is "both the same" — true when both are off or both are on.
• A < B = NOT A AND B. The mirror of the greater-than case.

Because the three conditions are mutually exclusive and cover every input, exactly one lamp is lit at any moment. Always. No matter what you do with the levers.

A	B	A>B	A=B	A<B
0	0	0	1	0
0	1	0	0	1
1	0	1	0	0
1	1	0	1	0

Why it matters

This is the building block for any magnitude comparator. To compare multi-bit numbers, you chain these from the most significant bit down: if the top bits differ, that's the answer; if they're equal, fall through to the next bit. The same "is A greater, equal, or less" cascade shows up in sorting networks, priority encoders, and every if (a > b) your CPU has ever evaluated.

Try this

• Try all four combinations of the levers. Notice that moving between them always swaps which single lamp is lit — there's no transitional state with two lamps on.
• Cover the middle lamp with your hand. The outer two lamps alone tell you whether A and B differ — useful as an equality detector built the long way around.