Universal Logic Gates
A universal logic gate is a logic gate that can be used to construct all other logic gates. There are many articles about how NAND and NOR are universal gates, but many of these articles omit other gates that are also universal gates. This article covers two input logic gates, demonstrates that the NAND gate is a universal gate, and demonstrates how other gates are universal gates that can be used to construct any logic gate.
For 2 logical inputs, there are 16 possible logic gates. Table 1 defines the 16 logic gates, shows logical symbol, behavior in symbolic logic, provides name, notes.
People are most familiar with the ones that are named: NAND, NOR, AND, OR, XOR. Table 1 includes these, as well as AND and OR gates with either input negated, logic 0 and logic 1(which aren't really logic gates), and some gates that aren't really binary.
Table 1: 2 Input Logic Gates
NAND gates can be connected to form any other logic gates. Figures 1,2,3 show how NAND gates can be connected to form INVERTER, AND, and OR gates. These gates can be combined to form the other logic gates according to the symbolic logic definitions in table 1.
Gates 2,4,11, and 13 are universal gates. To prove this, we demonstrate building NAND gates with these gates. Gates 7 and 8 are the same gate but with different inputs inverted. Gates 9 and 10 are the same gate but with different inputs inverted. Proving that gates 2 and 11 are universal is enough to prove that gate 4 and 13 are universal as well.
Gate 11: A | ~B
Figures 4 and 5 show show gate 11 can be combined to create INVERTER and AND gates. INVERTER and AND gates can be connected in series to construct a NAND gate. Because gate 11 can be used to construct a NAND gate, and a NAND gate is a universal gate, gate 11 is a universal gate. Since gates 11 and 13 are mirrors of each other, gate 13 is also a universal gate.
Gate 2: A & ~B
Figures 6 and 7 show show gate 2 can be combined to create INVERTER and AND gates. INVERTER and AND gates can be connected in series to construct a NAND gate. Because gate 2 can be used to construct a NAND gate, and a NAND gate is a universal gate, gate 2 is a universal gate. Since gates 2 and 4 are mirrors of each other, gate 4 is also a universal gate.
