NOT from NAND: NOT A = NAND(A, A)
NOT from NAND: an inverter is a NAND gate with both inputs fed the same signal, NOT A = NAND(A, A).
NOT is built from one NAND gate whose two inputs receive the same value.
NAND(A, A) is NOT (A AND A), and A AND A is A by idempotence, so the gate outputs NOT A. This is the first of the three constructions that make NAND universal: with an inverter in hand, AND is a NAND followed by this inverter, and OR is a NAND of two inverted inputs.
The engine's NAND-only converter uses this construction by default. It also offers the alternative NAND(A, TRUE), which ties the second input to a constant instead of duplicating the signal; the two are equivalent, and the converter switches to the constant form only when duplicating a large subtree would exceed its size bound.
The two-row proof is below, and the diagram shows the gate with its fan-in tied together.
Statement
NOT A = NAND(A, A) Proof by truth table: NOT A = NAND(A, A)
| # | A | lhs | rhs |
|---|---|---|---|
| 0 | FALSE | TRUE | TRUE |
| 1 | TRUE | FALSE | FALSE |
The same law in four notations
NOT A!A¬AA'\overline{A}NAND(A, A)NAND(A, A)NAND(A, A)NAND(A, A)\overline{A \cdot A}The common slip
Feeding the two inputs different signals. NAND(A, B) is not an inverter of anything; only the tied form inverts.
Questions
Why not just use a NOT gate?
In a chip built from one gate type there is no NOT gate; every function, inverters included, has to be made from the available gate. The construction is how a NAND-only family gets its inverter.
Related
Lesson that teaches this
Tools
- Boolean algebra calculator and simplifier with steps (every law named and verified)
- Logical equivalence checker: two expressions, exact verdict, counterexample
Practice
See also
- AND from NAND gates: A AND B = NAND(NAND(A, B), NAND(A, B))
- OR from NAND gates: A OR B = NAND(NAND(A, A), NAND(B, B))
- NAND elimination: NAND(A, B) = NOT (A AND B)
- NAND-only and NOR-only converter with equivalence proof
Laws
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