# NOR gate

Gate types
NOT
AND NAND
OR NOR
XOR XNOR

A NOR gate (from English : n ot or - not or , or from English nor - (neither -) nor; also called Peirce function after Charles S. Peirce ) is a logic gate with two or more inputs A, B, ... and an output Y, between which there is a logical NOT OR link . A NOR gate outputs 1 (w) when all inputs are 0 (f). In all other cases, i. H. if at least one input is 1, a 0 is output.

The literature uses the following notations for the NOR operation of variables A and B :

${\ displaystyle A \, \ operatorname {NOR} \, B \ qquad A \ downarrow B \ qquad \ neg \ left (A \ lor B \ right) \ qquad A \; \; \! \! {\ overline {\ lor}} \; \; \! \! B \ qquad {\ overline {A \ lor B}} \ qquad {\ overline {A + B}} \ qquad A \; \; \! \! {\ overline { +}} \; \; \! \! B \ qquad \ neg \ left (A + B \ right)}$

## Overview

function Circuit symbol Truth table Relay logic
IEC 60617-12 US ANSI 91-1984 DIN 40700 (before 1976)
${\ displaystyle Y = {\ overline {A \ vee B}}}$

${\ displaystyle Y = A \; \; \! \! {\ overline {\ vee}} \; \; \! \! B}$

${\ displaystyle Y = {\ overline {A + B}}}$

${\ displaystyle Y = A \ downarrow B}$

${\ displaystyle Y = A \ backslash B}$
A. B. Y = A ⊽ B
0 0 1
0 1 0
1 0 0
1 1 0

## realization

The electronic implementation takes place, for example (with positive logic ) with two (or correspondingly more) switches ( transistors ) connected in parallel , which connect output Q to ground (logic 0) as soon as one of them is switched on. If all are off, the ground connection is interrupted and the output Q is at positive potential (logical 1).

## Logic synthesis

According to the following logical equivalence , a NOR operation can also be built up from NAND gates alone :

${\ displaystyle x \; \; \! \! {\ overline {\ lor}} \; \; \! \! y = \ left [\ left (x \; \; \! \! {\ overline {\ land}} \; \; \! \! x \ right) \; \; \! \! {\ overline {\ land}} \; \; \! \! \ left (y \; \; \! \ ! {\ overline {\ land}} \; \; \! \! y \ right) \ right] \; \; \! \! {\ overline {\ land}} \; \; \! \! \ left [\ left (x \; \; \! \! {\ overline {\ land}} \; \; \! \! x \ right) \; \; \! \! {\ overline {\ land}} \ ; \; \! \! \ left (y \; \; \! \! {\ overline {\ land}} \; \; \! \! y \ right) \ right]}$

Logical links and their implementation using NOR gates:

With the Peirce function alone, all two-valued truth functions can be represented, i.e. every Boolean function is equivalent to a formula that only contains the NOR function. Because of this property of functional completeness , the peirce function is called a basis of the two-digit logical functions (another basis is the NAND function).

 NOT ( negation , not) ${\ displaystyle {\ overline {x}}}$ ${\ displaystyle \ equiv}$ ${\ displaystyle x \; \; \! \! {\ overline {\ lor}} \; \; \! \! x}$ AND ( conjunction , and) ${\ displaystyle x \ land y}$ ${\ displaystyle \ equiv}$ ${\ displaystyle \ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! x \ right) \; \; \! \! {\ overline {\ lor} } \; \; \! \! \ left (y \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right)}$ NAND (not-and) ${\ displaystyle x {\ overline {\ land}} y}$ ${\ displaystyle \ equiv}$ ${\ displaystyle \ left [\ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! x \ right) \; \; \! \! {\ overline { \ lor}} \; \; \! \! \ left (y \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right) \ right] \; \ ; \! \! {\ overline {\ lor}} \; \; \! \! \ left [\ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! x \ right) \; \; \! \! {\ overline {\ lor}} \; \; \! \! \ left (y \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right) \ right]}$ OR ( disjunction , or) ${\ displaystyle x \ lor y}$ ${\ displaystyle \ equiv}$ ${\ displaystyle \ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right) \; \; \! \! {\ overline {\ lor} } \; \; \! \! \ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right)}$ NOR (not-or) ${\ displaystyle x \; \; \! \! {\ overline {\ lor}} \; \; \! \! y}$ ${\ displaystyle \ equiv}$ ${\ displaystyle x \; \; \! \! {\ overline {\ lor}} \; \; \! \! y}$ XOR (Exclusive-Or) ${\ displaystyle x \; \; \! \! {\ underline {\ lor}} \; \; \! \! y}$ ${\ displaystyle \ equiv}$ ${\ displaystyle \ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right) \; \; \! \! {\ overline {\ lor} } \; \; \! \! \ left [\ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! x \ right) \; \; \! \! {\ overline {\ lor}} \; \; \! \! \ left (y \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right) \ right]}$ XNOR (Exclusive-Not-Or) ${\ displaystyle x \; \; \! \! {\ overline {\ underline {\ lor}}} \; \; \! \! y}$ ${\ displaystyle \ equiv}$ ${\ displaystyle \ left [\ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right) \; \; \! \! {\ overline { \ lor}} \; \; \! \! x \ right] \; \; \! \! {\ overline {\ lor}} \; \; \! \! \ left [\ left (x \; \ ; \! \! {\ overline {\ lor}} \; \; \! \! y \ right) \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right]}$ implication ${\ displaystyle x \ rightarrow y}$ ${\ displaystyle \ equiv}$ ${\ displaystyle \ left [\ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! x \ right) \; \; \! \! {\ overline { \ lor}} \; \; \! \! y \ right] \; \; \! \! {\ overline {\ lor}} \; \; \! \! \ left [\ left (x \; \ ; \! \! {\ overline {\ lor}} \; \; \! \! x \ right) \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right]}$ ${\ displaystyle x \ leftarrow y}$ ${\ displaystyle \ equiv}$ ${\ displaystyle \ left [x \; \; \! \! {\ overline {\ lor}} \; \; \! \! \ left (y \; \; \! \! {\ overline {\ lor} } \; \; \! \! y \ right) \ right] \; \; \! \! {\ overline {\ lor}} \; \; \! \! \ left [x \; \; \! \! {\ overline {\ lor}} \; \; \! \! \ left (y \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right) \ right]}$ equivalence ${\ displaystyle x \ leftrightarrow y}$ ${\ displaystyle \ equiv}$ ${\ displaystyle \ left [\ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right) \; \; \! \! {\ overline { \ lor}} \; \; \! \! x \ right] \; \; \! \! {\ overline {\ lor}} \; \; \! \! \ left [\ left (x \; \ ; \! \! {\ overline {\ lor}} \; \; \! \! y \ right) \; \; \! \! {\ overline {\ lor}} \; \; \! \! y \ right]}$ Verum (always true) ${\ displaystyle \ top}$ ${\ displaystyle \ equiv}$ ${\ displaystyle \ left [\ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! x \ right) \; \; \! \! {\ overline { \ lor}} \; \; \! \! x \ right] \; \; \! \! {\ overline {\ lor}} \; \; \! \! \ left [\ left (x \; \ ; \! \! {\ overline {\ lor}} \; \; \! \! x \ right) \; \; \! \! {\ overline {\ lor}} \; \; \! \! x \ right]}$ Falsum (always wrong) ${\ displaystyle \ bot}$ ${\ displaystyle \ equiv}$ ${\ displaystyle \ left (x \; \; \! \! {\ overline {\ lor}} \; \; \! \! x \ right) \; \; \! \! {\ overline {\ lor} } \; \; \! \! x}$

## literature

• Ulrich Tietze, Christoph Schenk: Semiconductor circuit technology . 12th edition. Springer, 2002, ISBN 3-540-42849-6 .