PYTHON · NUMPY

The Universal Gate

Build AND, OR, and NOT using only NAND — proving one gate is enough to compute anything.

01Challenge

A fab really does etch millions of copies of a few simple gates. Prove the deepest fact behind that: from NAND alone, reconstruct AND, OR, and NOT so every truth table checks out.

Predict firstHow many DIFFERENT kinds of logic gate do you need to build any digital circuit — an entire CPU?

02Model

A chip is an ocean of tiny switches, wired together. The deepest fact in digital design is that a single gate, NAND, is universal: any Boolean function whatsoever can be built from NANDs alone. NOT is a NAND wired to itself; AND is a NAND then a NOT; OR is a NAND of two NOTs (De Morgan). Once you have those, you have all of logic — and everything above, from an adder to a neural accelerator, is just more of the same.

03Guided practice

Worked: define NAND(a,b)=1-(a&b). Faded: build NOT from one NAND, AND from two, OR from three. Independent: verify all four input pairs for each gate.

04Feedback

PASS WHEN You reconstructed all of Boolean logic from a single gate — the reason a fab only needs to master a handful of devices.

If your run fails, check:

FAIL: AND or OR is off — remember AND = NOT(NAND), and OR uses De Morgan: NAND(NOT a, NOT b).
FAIL: NOT is wrong — a NAND with both inputs tied together inverts.

05Retrieve & space

Why does universality matter to a fab? (You only need to manufacture one gate well to build any logic.)
What is NOT in terms of NAND? (NAND(a,a) — both inputs tied together.)
Which law turns OR into NANDs? (De Morgan: a OR b = NOT(NOT a AND NOT b).)

06Mastery & project

Builds AND/OR/NOT from NAND and verifies each truth table (L1: functional completeness of a universal gate).

These gates are the primitives every later lesson composes — the adder, the MAC, and the accelerator are all NANDs underneath.

← Silicon for Physical AI The Adder →

PYTHON · NUMPY

The Universal Gate

Rebuild the basic gates from NAND alone and check every truth table.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you