/* MINDFIRE BLOG */

Last time, in The Atomization of Thought, we saw how AND, OR, and NOT join forces to build every imaginable Boolean function. Today, we’re going a step further: we’ll collapse those three operations into one—the mighty NAND gate—and see why hardware engineers can’t stop using it.

From Three to Two: Borrowing OR via Duality

Imagine you only own AND and NOT gates. You still need OR, right? Not really. Thanks to De Morgan, OR simply emerges when you flip everything twice:

A ∨ B is nothing more than ¬(¬A ∧ ¬B).

First negate A and B, then feed them into an AND, then negate the result. Voilà—OR, at no extra cost. Now our toolkit has shrunk from three operations down to two: AND and NOT.

One Gate to Rule Them All: NAND

What if we could combine AND and NOT into a single device? Enter the NAND gate, defined as:

A NAND B = ¬(A ∧ B)

It does both jobs at once: it computes A∧B and flips it. Here’s its truth table:

Because a single NAND gate already includes negation, we can recreate the two-gate toolkit using only NANDs. Want NOT A? Just wire A into both inputs of one NAND. Need A∧B? NAND followed by another NAND will do the trick: first take A NAND B, then NAND that result with itself. Even OR falls neatly into place: NAND the negations of each input.

Why Silicon Prefers NAND

In CMOS technology, NAND gates are cheaper to build and switch faster than NOR gates. That’s why most chips—adders, multiplexers, flip-flops—all ultimately boil down to interconnected NANDs. By standardizing on one gate type, manufacturers reduce complexity, shrink die size, and boost yield.

The Journey from Three to One

We started with three primitives—AND, OR, NOT—then used duality to derive OR from AND and NOT. Finally, we realized that AND and NOT can live inside a single NAND gate. In just three steps, we’ve distilled the essence of Boolean logic into a lone operator, yet retained the power to construct any function in the digital universe.