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Booleans

represents the domain of Booleans, as in x∈Booleans.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Possible Issues  
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Booleans

Booleans

represents the domain of Booleans, as in x∈Booleans.

Details

  • The domain of Booleans is taken to consist of the symbols True and False.
  • x∈Booleans evaluates immediately if x is explicitly True or False.
  • Simplify[expr∈Booleans] can be used to try to determine whether an expression is Boolean, with no undetermined variables.
  • Booleans is output in TraditionalForm as TemplateBox[{}, Booleans]. This typeset form can be input using bools.

Examples

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Basic Examples  (2)

Constants and variables:

Wolfram Language code: Element[True, Booleans]
Wolfram Language code: Element[x, Booleans]

Solution domain:

Wolfram Language code: FindInstance[Xor[x, y], {x, y}, Booleans]

Scope  (3)

Domain for FindInstance:

Wolfram Language code: FindInstance[p && !(!p || !q), {p, q}, Booleans]
Wolfram Language code: FindInstance[a && x < y && Element[a, Booleans], {a, x, y}]

Domain for Resolve, in this case solving satisfiability:

Wolfram Language code: Resolve[Exists[{p, q}, p || q && !q], Booleans]

TraditionalForm for formatting:

Wolfram Language code: Booleans // TraditionalForm

Applications  (1)

Use Simplify to determine whether an expression is Boolean, with no undetermined variables:

Wolfram Language code: Simplify[n^2 < EulerPhi[n]^3∈Booleans, n > 42 && n∈Integers]

Possible Issues  (1)

0 and 1 are not used to stand for False and True:

Wolfram Language code: Element[0, Booleans]
Wolfram Language code: Element[1, Booleans]
Wolfram Research (1999), Booleans, Wolfram Language function, https://reference.wolfram.com/language/ref/Booleans.html (updated 2017).

Text

Wolfram Research (1999), Booleans, Wolfram Language function, https://reference.wolfram.com/language/ref/Booleans.html (updated 2017).

CMS

Wolfram Language. 1999. "Booleans." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/Booleans.html.

APA

Wolfram Language. (1999). Booleans. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Booleans.html

BibTeX

@misc{reference.wolfram_2026_booleans, author="Wolfram Research", title="{Booleans}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/Booleans.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_booleans, organization={Wolfram Research}, title={Booleans}, year={2017}, url={https://reference.wolfram.com/language/ref/Booleans.html}, note=[Accessed: 13-June-2026]}