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Xor
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Xor

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Xor[e1,e2,…]

is the logical XOR (exclusive OR) function. It gives True if an odd number of the e_(i) are True, and the rest are False. It gives False if an even number of the e_(i) are True, and the rest are False.

Details

  • Xor[e1,e2,…] can be input in StandardForm and InputForm as e_(1) xor e_(2) xor .... The character ⊻ can be entered as xor or \[Xor].
  • Xor gives symbolic results when necessary, applying various simplification rules to them.
  • Unlike And, Nand, Or, and Nor, Xor must always test all its arguments, and so is not a control structure, and does not have attribute HoldAll.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

In[1]:=1
✖
https://wolfram.com/xid/02c7m-l5bvc9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02c7m-dev8m0
Out[2]=2

Enter using xor:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-cyw7h8
Out[1]=1

Scope  (4)Survey of the scope of standard use cases

Xor is associative and commutative:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-e12p2h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02c7m-eily5h
Out[2]=2

Do symbolic simplification:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-b4vw74
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02c7m-bzysuh
Out[2]=2

Expand in terms of And, Or, and Not:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-bnrg9g
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-4vzw0

Applications  (3)Sample problems that can be solved with this function

Find the Xor of two regions in 2D:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-p4rn8i
Out[1]=1

Find the Xor of three regions in 3D:

In[2]:=2
✖
https://wolfram.com/xid/02c7m-cor3n
Out[2]=2

A cellular automaton based on Xor:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-gddzdp
Out[1]=1

Find the area of the symmetric difference of sets given by algebraic conditions:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-fe4ca9
Out[1]=1

This shows the set:

In[2]:=2
✖
https://wolfram.com/xid/02c7m-c14dt7
Out[2]=2

Properties & Relations  (3)Properties of the function, and connections to other functions

Truth table for binary Xor:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-ib02ro
Out[1]=1

Ternary Xor:

In[2]:=2
✖
https://wolfram.com/xid/02c7m-cwl4xo
Out[2]=2

Use BooleanConvert to expand in terms of And, Or, and Not:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-cz4euq
Out[1]=1

Xor of conditions in Boole functions:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-efrdg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02c7m-hzp6na
Out[2]=2

Neat Examples  (2)Surprising or curious use cases

The Xor of disks on a circle:

In[1]:=1
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https://wolfram.com/xid/02c7m-d221dv
In[2]:=2
✖
https://wolfram.com/xid/02c7m-lusa3y

Generate three disks on a circle:

In[3]:=3
✖
https://wolfram.com/xid/02c7m-kslt4
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02c7m-mfkrzw
Out[4]=4

A truth table for a 12-variable Xor function:

In[1]:=1
✖
https://wolfram.com/xid/02c7m-hdw292
Out[1]=1
Wolfram Research (1988), Xor, Wolfram Language function, https://reference.wolfram.com/language/ref/Xor.html (updated 2003).
✖
Wolfram Research (1988), Xor, Wolfram Language function, https://reference.wolfram.com/language/ref/Xor.html (updated 2003).

Text

Wolfram Research (1988), Xor, Wolfram Language function, https://reference.wolfram.com/language/ref/Xor.html (updated 2003).

✖
Wolfram Research (1988), Xor, Wolfram Language function, https://reference.wolfram.com/language/ref/Xor.html (updated 2003).

CMS

Wolfram Language. 1988. "Xor." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/Xor.html.

✖
Wolfram Language. 1988. "Xor." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/Xor.html.

APA

Wolfram Language. (1988). Xor. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Xor.html

✖
Wolfram Language. (1988). Xor. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Xor.html

BibTeX

@misc{reference.wolfram_2025_xor, author="Wolfram Research", title="{Xor}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/Xor.html}", note=[Accessed: 20-June-2025 ]}

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@misc{reference.wolfram_2025_xor, author="Wolfram Research", title="{Xor}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/Xor.html}", note=[Accessed: 20-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_xor, organization={Wolfram Research}, title={Xor}, year={2003}, url={https://reference.wolfram.com/language/ref/Xor.html}, note=[Accessed: 20-June-2025 ]}

✖
@online{reference.wolfram_2025_xor, organization={Wolfram Research}, title={Xor}, year={2003}, url={https://reference.wolfram.com/language/ref/Xor.html}, note=[Accessed: 20-June-2025 ]}