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Each of the singleton sets { NAND} and { NOR} is functionally complete. However, the set { AND, OR} is incomplete, due to its inability to express NOT. A gate (or set of gates) that is functionally complete can also be called a universal gate (or a universal set of gates).
A single NOR gate. A NOR gate or a NOT OR gate is a logic gate which gives a positive output only when both inputs are negative.. Like NAND gates, NOR gates are so-called "universal gates" that can be combined to form any other kind of logic gate.
A NAND gate is equivalent to an OR gate with negated inputs, and a NOR gate is equivalent to an AND gate with negated inputs. This leads to an alternative set of symbols for basic gates that use the opposite core symbol (AND or OR) but with the inputs and outputs negated. Use of these alternative symbols can make logic circuit diagrams much ...
A CMOS transistor NAND element. V dd denotes positive voltage.. In CMOS logic, if both of the A and B inputs are high, then both the NMOS transistors (bottom half of the diagram) will conduct, neither of the PMOS transistors (top half) will conduct, and a conductive path will be established between the output and Vss (ground), bringing the output low.
The NAND gate has the property of functional completeness, which it shares with the NOR gate. That is, any other logic function (AND, OR, etc.) can be implemented using only NAND gates. [2] An entire processor can be created using NAND gates alone. In TTL ICs using multiple-emitter transistors, it also requires fewer transistors than a NOR gate.
The NOR gate is a digital logic gate that implements logical NOR - it behaves according to the truth table to the right. A HIGH output (1) results if both the inputs to the gate are LOW (0); if one or both input is HIGH (1), a LOW output (0) results.
In fact, both NAND and NOR gates are so-called "universal gates" and any logical function can be constructed from either NAND logic or NOR logic alone. If the four NAND gates are replaced by NOR gates, this results in an XNOR gate, which can be converted to an XOR gate by inverting the output or one of the inputs (e.g. with a fifth NOR gate).
Universal Logic is an emerging interdisciplinary field involving logic, non-classical logic, categorical logic, set theory, foundation of logic, and the philosophy and history of logic. The goal of the field is to develop an understanding of the nature of different types of logic.