Gate Universality: Replicating Any Logic Function with NAND or NOR Gates

NAND and NOR gates are the only standard logic families that can, by themselves, implement any other logic operation when combined in sufficient numbers. This universal property is the foundation of many digital control systems that rely exclusively on one of these gate types.

For instance, a simple OR function can be built from just three NAND gates, and the same can be achieved with NOR gates. The following sections illustrate how every basic gate—NOT, AND, OR, NAND, and NOR—can be constructed using only NANDs or only NORs.

Constructing the NOT Function

Both NAND and NOR gates can serve as inverters. The most practical approach for TTL devices is to tie the unused input of a NAND gate to the supply rail (+V), while for NOR gates the unused input is tied to ground. This method minimizes current loading on the driving gate (TTL) and reduces input capacitance (CMOS). The alternative—joining the two inputs together—works logically but is less efficient from a hardware perspective.

Constructing the “Buffer” Function

A buffer can be realized by cascading two inverter stages. The output then mirrors the input exactly, preserving the logical level without inversion.

Constructing the AND Function

To obtain an AND from NAND gates, simply add an inverter to the NAND output. This effectively removes the NAND’s inherent NOT operation. With NOR gates, the same AND behavior is achieved by inverting all inputs before feeding them into the NOR.

Constructing the NAND Function

A NAND gate is already available; no additional circuitry is required. To make a NOR gate emulate NAND, invert every input and the output of the NOR gate. For a two‑input NOR, this demands three extra NOR inverters.

Constructing the OR Function

Inverting the output of a NOR gate produces an OR function. Conversely, a NAND gate can emulate OR by inverting all of its inputs. This reciprocal relationship is a direct application of De Morgan’s Theorem, which is fundamental to Boolean algebra.

Constructing the NOR Function

To use a NAND gate as a NOR, invert all of its inputs and its output. This mirrors the process used to make a NOR emulate NAND.

Review:

• Both NAND and NOR gates are universal: they can replicate any logic function when properly interconnected.

Related Worksheets:

• TTL Logic Gates Worksheet
• Boolean Algebra Worksheet