NOR gate

A NOR gate outputs the complement of an OR: $1$ only when all inputs are $0$.

$f=\overline{x_1+x_2}$

$x_1$	$x_2$	$\overline{x_1+x_2}$
0	0	1
0	1	0
1	0	0
1	1	0

Drawn as an OR symbol with a bubble at the output.

Image: NOR gate symbol, Public domain — distinctive-shape NOR gate (ANSI/IEEE 91-1984): OR body with an output bubble.

CMOS implementation

A 2-input NOR mirrors a NAND gate structurally but with the networks swapped: 2 PMOS in series in the pull-up, 2 NMOS in parallel in the pull-down.

The pull-up chain conducts only when both PMOS are on (both inputs $0$), pulling the output high. For any other combination, at least one NMOS in the pull-down is on, dragging the output low.

In silicon NOR is slower than NAND for the same logic depth, because the series PMOS chain has more resistance than a series NMOS chain (holes are slower charge carriers than electrons). When designers have a choice, they prefer NAND-based logic. NOR is the better fit when the natural expression of a function lends itself to it, like the cross-coupled latch, which is built from two NORs.

Universality

NOR is also a universal gate — any Boolean function can be built from NOR alone, just like NAND.

Conversions:

• NOT: $\overline{x+x}=\overline{x}$.
• OR: NOR followed by NOR-as-NOT: $(x\text{ NOR }y)\text{ NOR }(x\text{ NOR }y)=x+y$.
• AND: invert both inputs via NOR-as-NOT, then NOR: $(x\text{ NOR }x)\text{ NOR }(y\text{ NOR }y)=\overline{\overline{x}+\overline{y}}=xy$ by De Morgan’s Laws.

NOR-NOR synthesis

Dual to NAND-NAND synthesis. Convert a POS expression to an all-NOR implementation:

1. Write $f$ in POS form.
2. Apply double negation and push the inner negation into the AND using De Morgan’s Laws. Each OR term becomes the NOR of its literals, and the outer AND becomes a final NOR.

Useful identities:

• $a\cdot b=\overline{\overline{a}+\overline{b}}=\overline{a}\text{ NOR }\overline{b}$ — AND by NOR-ing the inverted inputs (De Morgan).
• $a+b=\overline{a\text{ NOR }b}=(a\text{ NOR }b)\text{ NOR }(a\text{ NOR }b)$ — recover an OR by inverting a NOR.
• $a\cdot b=(a\text{ NOR }a)\text{ NOR }(b\text{ NOR }b)$ — AND built entirely from NOR, using NOR-as-NOT to invert each input first.

Pick NAND-NAND for SOP-style designs and NOR-NOR for POS-style designs. Either works for any function; the difference is which form has fewer terms after minimization.