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Xnor[e1,e2,…]

is the logical XNOR (not XOR) function. It gives True if an even number of the ei are True, and the rest are False. It gives False if an odd number of the ei are True, and the rest are False.

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Xnor

Xnor[e1,e2,…]

is the logical XNOR (not XOR) function. It gives True if an even number of the ei are True, and the rest are False. It gives False if an odd number of the ei are True, and the rest are False.

Details

  • Xnor[e1,e2,…] can be input in StandardForm and InputForm as eie2…. The character can be entered as xnor or \[Xnor].
  • Xnor gives symbolic results, applying various simplification rules automatically.
  • Unlike And, Nand, Or, and Nor, Xnor must always test all its arguments, so is not a control structure, and does not have attribute HoldAll.

Examples

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Basic Examples  (3)

Wolfram Language code: Xnor[True, False]
Wolfram Language code: Xnor[True, True]

Xnor in disjunctive normal form:

Wolfram Language code: BooleanConvert[Xnor[a, b]]

Enter using xnor:

Wolfram Language code: ab

Scope  (1)

Automatic simplifications:

Wolfram Language code: Xnor[]
Wolfram Language code: Xnor[a]

Elimination of constants:

Wolfram Language code: Xnor[a, True, b]
Wolfram Language code: Xnor[a, False, b]

Xnor is commutative and sorts its arguments:

Wolfram Language code: Xnor[c, a, b]

Properties & Relations  (5)

Truth table for binary Xnor:

Wolfram Language code: BooleanTable[{x, y, Xnor[x, y]}, {x, y}]//Grid

Ternary Xnor:

Wolfram Language code: BooleanTable[{x, y, z, Xnor[x, y, z]}, {x, y, z}]//Grid

Xnor in variables has a disjunctive normal form of length :

Wolfram Language code: BooleanConvert[Xnor@@Array[a, 2]]
Wolfram Language code: BooleanConvert[Xnor@@Array[a, 3]]
Wolfram Language code: Table[Length[BooleanConvert[Xnor@@Array[a, n]]], {n, 2, 10}]

Xnor is equivalent to the negation of Xor:

Wolfram Language code: TautologyQ[Equivalent[Xnor[a, b, c], ¬Xor[a, b, c]]]
Wolfram Language code: TautologyQ[Equivalent[Xnor@@Array[a, 10], Not[Xor@@Array[a, 10]]]]

For two variables Xnor is the same as Equivalent:

Wolfram Language code: TautologyQ[Equivalent[Xnor[a, b], Equivalent[a, b]]]

But they differ for more than two variables:

Wolfram Language code: TautologyQ[Equivalent[Xnor[a, b, c], Equivalent[a, b, c]]]

Xnor is a special case of BooleanCountingFunction:

Wolfram Language code: xnor[v__] := BooleanCountingFunction[{0, Length[{v}], 2}, Length[{v}]][v]
Wolfram Language code: TautologyQ[Equivalent[Xnor@@Array[a, 10], xnor@@Array[a, 10]]]

Neat Examples  (1)

The 2D truth table for Xnor of 12 variables:

Wolfram Language code: ArrayPlot[BooleanTable[Xnor@@Array[x, 12], Array[x, 6], Array[x, 6, 7]], ColorRules -> {False -> White, True -> Black}]
Wolfram Research (2008), Xnor, Wolfram Language function, https://reference.wolfram.com/language/ref/Xnor.html.

Text

Wolfram Research (2008), Xnor, Wolfram Language function, https://reference.wolfram.com/language/ref/Xnor.html.

CMS

Wolfram Language. 2008. "Xnor." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Xnor.html.

APA

Wolfram Language. (2008). Xnor. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Xnor.html

BibTeX

@misc{reference.wolfram_2026_xnor, author="Wolfram Research", title="{Xnor}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Xnor.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_xnor, organization={Wolfram Research}, title={Xnor}, year={2008}, url={https://reference.wolfram.com/language/ref/Xnor.html}, note=[Accessed: 12-June-2026]}