Digital Design Fundamentals·Signals, Bits & Logic
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Add an inversion bubble to a gate's output and you get its compound twin:
The little bubble on a symbol always means "invert this". Spotting it tells you half of what a gate does at a glance.
Figure: Gate symbols for NAND, NOR and XOR side by side: the AND and OR shapes each with a small inversion bubble on the output
Add an inversion bubble to a gate's output and you get its compound twin:
The little bubble on a symbol always means "invert this". Spotting it tells you half of what a gate does at a glance.
The circuit above feeds inputs a and b into a NAND, a NOR and an XOR gate at once. Toggle them and check each output against the table below.
| A | B | NAND | NOR | XOR |
|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 1 | 0 | 0 | 0 |
Compare the NAND column with AND from the last lesson: every row is flipped. NOR does the same to OR. XOR is the odd one out, a "these two disagree" detector, and it turns up all over binary arithmetic.
NAND is universal: any logic function at all, however large, can be built from NAND gates alone.
Figure: Three small schematics built only from NAND gates, each labelled with its gate count: NOT as one NAND with both inputs tied together
Once you can build NOT, AND and OR, you can build anything, so NAND alone is enough for an entire chip. NOR is universal too, in mirror image.
A NAND gate is the cheapest useful logic that CMOS can make: a handful of transistors, fast and compact. So the industry perfected it and built much of everything else from it.
