View source for XNOR gate

{{Short description|Digital logic gate}}
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{| class="wikitable floatright" style="text-align:center"
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! colspan="3" | XNOR gate [[truth table]]
|- bgcolor="#ddeeff"
|colspan=2|'''Input''' || '''Output'''
|- bgcolor="#ddeeff"
| A || B || A XNOR B
|-
|{{no2|0}} || {{no2|0}} || {{yes2|1}}
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|{{no2|0}} || {{yes2|1}} || {{no2|0}}
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|{{yes2|1}} || {{no2|0}} || {{no2|0}}
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|{{yes2|1}} || {{yes2|1}} || {{yes2|1}}
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{{Logical connectives sidebar}}

The '''XNOR gate''' (sometimes '''ENOR''', '''EXNOR''', '''NXOR''', '''XAND''' and pronounced as '''Exclusive NOR''') is a digital [[logic gate]] whose function is the [[logical complement]] of the Exclusive OR ([[XOR gate|XOR]]) gate.<ref name=":0">{{cite web|title=Exclusive-NOR Gate Tutorial|date=22 August 2013 |url=https://www.electronics-tutorials.ws/logic/logic_8.html|access-date=6 May 2018}}</ref> It is equivalent to the logical connective (<math>\leftrightarrow</math>) from [[mathematical logic]], also known as the material biconditional. The two-input version implements [[logical equality]], behaving according to the truth table to the right, and hence the gate is sometimes called an "equivalence gate". A high output (1) results if both of the inputs to the gate are the same. If one but not both inputs are high (1), a low output (0) results.

The [[Boolean algebra|algebraic notation]] used to represent the XNOR operation is <math>S = A \odot B</math>. The algebraic expressions <math>(A + \overline{B}) \cdot (\overline{A} + B)</math> and <math>A \cdot B + \overline A \cdot \overline B</math> both represent the XNOR gate with inputs ''A'' and ''B''.

==Symbols==
There are [[logic gate#Symbols|two symbols for XNOR gates]]: one with distinctive shape and one with rectangular shape and label.  Both symbols for the XNOR gate are that of the [[XOR gate]] with an added inversion bubble.

{{multiple image
| align = left
| direction = horizontal

| image1 = Xnor-gate-en.svg
| caption1 = Distinctive symbol
| width1 = {{#expr: (100 * 500 / 180) round 0}}

| image2 = IEC XNOR.svg
| caption2 = Rectangular symbol
| width2 = {{#expr: (100 * 128 / 74) round 0}}
}}
{{clear}}

==Hardware description==
XNOR gates are represented in most [[Transistor–transistor logic|TTL]] and [[CMOS]] [[Integrated circuit|IC]] families. The standard [[4000 series]] CMOS IC is the 4077, and the TTL IC is the 74266 (although an [[open-collector]] implementation). Both include four independent, two-input, XNOR gates. The (now obsolete) 74S135 implemented four two-input XOR/XNOR gates or two three-input XNOR gates. 

Both the TTL [[74LS]] implementation, the 74LS266, as well as the CMOS gates (CD4077, 74HC4077 and 74HC266 and so on) are available from most semiconductor manufacturers such as [[Texas Instruments]] or [[NXP]], etc.<ref>{{cite web|title=XNOR Logic Gates|url=https://mouser.com/Semiconductors/Logic-ICs/Logic-Gates/_/N-581zn?P=1yrp5qe|access-date=6 May 2018}}</ref> They are usually available in both through-hole [[Dual in-line package|DIP]] and [[small-outline integrated circuit|SOIC]] formats (SOIC-14, SOC-14 or TSSOP-14). 

Datasheets are readily available in most [[Datasheet#Datasheet Search Engines|datasheet databases]] and suppliers. 

==Implementation==
===AND-OR-Invert logic===
An XNOR gate can be implemented using a NAND gate and an [[OR-AND-Invert]] gate, as shown in the following picture. <ref>{{Cite web |url=https://sus.ziti.uni-heidelberg.de/Lehre/WS1617_DST/DST_Fischer_03_Logik_Gatter.pptx.pdf |title=Aussagenlogik und Gatter |access-date=2024-01-21 |last=Fischer |first=P. |publisher=University of Heidelberg}}</ref>
This is based on the identity

<math> \overline{a \veebar b} \iff \left(a \overline{\and} b\right) \overline{\and} \left(a \lor b\right)</math>

An alternative, which is useful when inverted inputs are also available (for example from a [[Flip-flop (electronics)|flip-flop]]), uses a 2-2 [[AND-OR-Invert]] gate, shown on below on the right.
<gallery widths="250">
XNOR OAI gates.svg|XNOR implemention using a NAND and an OAI gate
XNOR AOI inverted inputs gates.svg|XNOR implementation using a 2-2-AOI gate with normal and inverted inputs
</gallery>
===CMOS===
CMOS implementations based on the OAI logic above can be realized with 10 [[transistors]], as shown below. The implementation which uses both normal and inverted inputs uses 8 transistors, or 12 if inverters have to be used.
<gallery widths="200" heights="300">
XNOR OAI CMOS.svg|An XNOR-gate in CMOS using a NAND and an OR-AND-Invert gate
XNOR AOI CMOS inverted inputs.svg|An XNOR gate in CMOS using both normal and inverted inputs
</gallery>

==Pinout==
Both the 4077 and 74x266 devices (SN74LS266, 74HC266, 74266, etc.) have the same pinout diagram, as follows:

[[Image:XNOR Pinout.png|thumb|left|350px]]
Pinout diagram of the 74HC266N, 74LS266 and CD4077 quad XNOR plastic [[dual in-line package]] 14-pin package ('''PDIP-14''') [[Integrated Circuit|ICs]].

{{ordered list
 | list_style=list-style-position: inside;
 | Input A1
 | Input B1
 | Output Q1 (high if and only if A1 and B1 have the same logic level)
 | Output Q2
 | Input B2
 | Input A2
 | V<sub>ss</sub> (GND) common power and signal ground pin
 | Input A3
 | Input B3
 | Output Q3
 | Output Q4
 | Input B4
 | Input A4
 | V<sub>dd</sub> for CMOS (V<sub>cc</sub> for TTL) positive power supply (see datasheets for acceptable voltage ranges)
}}

==Alternatives==
[[File:3 gate XNOR.svg|frame|right|XNOR gate circuit using three mixed gates]]
If a specific type of gate is not available, a circuit that implements the same function can be constructed from other available gates. A circuit implementing an XNOR function can be trivially constructed from an XOR gate followed by a NOT gate. If we consider the expression <math>(A + \overline{B}) \cdot (\overline{A} + B)</math>, we can construct an XNOR gate circuit directly using AND, OR and NOT gates. However, this approach requires five gates of three different kinds.

As alternative, if different gates are available we can apply [[Boolean algebra]] to transform <math>(A + \overline{B}) \cdot (\overline{A} + B)  \equiv (A \cdot B) + ( \overline A \cdot \overline B )</math> as stated above, and apply [[de Morgan's Law]] to the last term to get <math>(A \cdot B) + \overline {(A + B)} </math> which can be implemented using only three gates as shown on the right.

An XNOR gate circuit can be made from four NOR gates. 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 NOR gates are replaced by NAND gates, this results in an XOR gate, which can be converted to an XNOR gate by inverting the output or one of the inputs (e.g. with a fifth NAND gate). 

{|  style="margin:auto; text-align:center;"
|-
! width="250" |Desired gate!!width=150|NAND construction!!width=150|NOR construction
|-
|[[Image:XNOR ANSI Labelled.svg]]||[[File:XNOR from NAND.svg]]||[[File:XNOR from NOR.svg]]
|}

An alternative arrangement is of five NAND gates in a topology that emphasizes the construction of the function from <math>(A \cdot B) + (\overline{A} \cdot \overline{B})</math>, noting from [[de Morgan's Law]] that a NAND gate is an inverted-input OR gate. Another alternative arrangement is of five NOR gates in a topology that emphasizes the construction of the function from <math>(A + \overline{B}) \cdot (\overline{A} + B)</math>, noting from [[de Morgan's Law]] that a NOR gate is an inverted-input AND gate.

{|  style="margin:auto; text-align:center;"
|-
! width="250" |Desired gate!!width=150|NAND construction!!width=150|NOR construction
|-
|[[Image:XNOR ANSI Labelled.svg]]||[[File:XNOR from NAND 2.svg]]||[[File:XNOR from NOR 2.svg]]
|}

For the NAND constructions, the lower arrangement offers the advantage of a shorter propagation delay (the time delay between an input changing and the output changing). For the NOR constructions, the upper arrangement requires fewer gates.

From the opposite perspective, constructing other gates using only XNOR gates is possible though XNOR is not a fully [[Logic_gate#Universal_logic_gates|universal logic gate]].  NOT and XOR gates can be constructed this way.

==More than two inputs==
Although other gates (OR, NOR, AND, NAND) are available from manufacturers with three or more inputs per gate, this is not strictly true with XOR and XNOR gates. However, extending the concept of the [[binary operators|binary]] logical operation to three inputs, the SN74S135 with two shared "C" and four independent "A" and "B" inputs for its four outputs, was a device that followed the truth table:

{| class="wikitable" style="text-align:center; background-color: #ddffdd;" 
|-
! style="background-color: #ddeeff;" colspan=3 | Input
! style="background-color: #ddeeff;"           | Output
|-
! style="background-color: #ddeeff;" | A
! style="background-color: #ddeeff;" | B
! style="background-color: #ddeeff;" | C
! style="background-color: #ddeeff;" | Q
|-
| {{no2|0}} || {{no2|0}} || {{no2|0}} || {{yes2|1}}
|-          
| {{no2|0}} || {{no2|0}} || {{yes2|1}} || {{no2|0}}
|-          
| {{no2|0}} || {{yes2|1}} || {{no2|0}} || {{no2|0}}
|-          
| {{no2|0}} || {{yes2|1}} || {{yes2|1}} || {{yes2|1}}
|-          
| {{yes2|1}} || {{no2|0}} || {{no2|0}} || {{no2|0}}
|-          
| {{yes2|1}} || {{no2|0}} || {{yes2|1}} || {{yes2|1}}
|-          
| {{yes2|1}} || {{yes2|1}} || {{no2|0}} || {{yes2|1}}
|-          
| {{yes2|1}} || {{yes2|1}} || {{yes2|1}} || {{no2|0}}
|-          
         
|}

This is effectively Q = NOT ((A XOR B) XOR C). Another way to interpret this is that the output is true if an even number of inputs are true. It does not implement a logical "equivalence" function, unlike two-input XNOR gates.

==See also==
{{Commons category|XNOR gates}}
*[[AND gate]]
*[[OR gate]]
*[[Inverter (logic gate)|NOT gate]]
*[[NAND gate]]
*[[NOR gate]]
*[[XOR gate]]
*[[Kronecker delta|Kronecker delta function]]
*[[Logical biconditional]]
*[[If and only if]]

== References ==
<references/>

== External links ==
*[http://www.ti.com/lit/ds/snos367a/snos367a.pdf Texas Instruments© CD4077 Datasheet]
{{Logical connectives}}

{{DEFAULTSORT:Xnor Gate}}
[[Category:Logic gates]]

[[it:Algebra di Boole#XNOR]]