NAND / NOR universality
NAND and NOR are each universal: every Boolean function can be built from copies of one of them alone. The construction always starts with negation — x↑x = x′ and x↓x = x′ — and then De Morgan supplies the rest. This is why real fabrication processes standardise on NAND, and why "convert this circuit to NAND-only" is a standard exam task.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Build NOT firstTie both inputs together: x↑x = (x·x)′ = x′, and x↓x = (x + x)′ = x′.
- Build AND from NANDx·y = (x↑y)↑(x↑y) — one NAND then an inverting NAND.
- Build OR from NANDBy De Morgan, x + y = (x′·y′)′ = (x↑x)↑(y↑y).
- Convert whole circuits systematicallyPush bubbles through with De Morgan: an AND-OR two-level circuit becomes NAND-NAND with no change to the gate topology.
Worked example
Express x + y using only NOR gates (↓).
- x ↓ y = (x + y)′.
- So x + y is the complement of x ↓ y.
- Complement using NOR: (x ↓ y) ↓ (x ↓ y).
- Check: if x = 1, y = 0, then x↓y = 0 and 0↓0 = 1 ✓.
Answer. x + y = (x ↓ y) ↓ (x ↓ y) — two NOR gates.
Where marks get dropped
These are the specific errors that cost credit on nand / nor universality questions — QED's rubric penalises each of them separately.
- Mixing NAND and NOR when the question says one gate type only.
- Forgetting the final inversion. x↑y is NOT AND — an extra inverting stage is needed to get AND.
- Assuming the NAND-only circuit has the same gate count as the original. It usually has more, which is the trade-off for a uniform gate library.
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NAND / NOR universality — frequently asked questions
Why is NAND preferred in CMOS?
A CMOS NAND uses fewer transistors than a NOR of the same fan-in and has better performance because it stacks PMOS transistors in parallel rather than in series.
Are there other universal gates?
Any gate whose function is not truth-preserving, falsity-preserving, monotone, affine or self-dual is universal on its own — by Post’s theorem, NAND and NOR are the only universal binary gates.
Does AND-OR to NAND-NAND change the logic?
No. Inserting two bubbles on each internal wire cancels, so the two-level topology is preserved exactly — which is why the conversion is mechanical.
The rest of Boolean Algebra
Axioms, laws, simplification and Boolean functions. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Boolean axioms & laws
- 2The duality principle
- 3Simplifying Boolean expressions
- 4Boolean functions & truth tables
- 5Boolean algebra, logic & set algebra
- 6Sum-of-products & product-of-sums
- 7Karnaugh maps & minimal expressions
- 8Logic gates & translating circuits
- 9NAND / NOR universality
- 10Don’t-care conditions in minimisation
- 11Quine–McCluskey minimisation
- 12Half adders, full adders & multiplexers
- 13Shannon expansion & binary decision diagrams
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