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LogicalExpand[expr]

expands out logical combinations of equations, inequalities, and other functions.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Neat Examples  
See Also
Tech Notes
Related Guides
History
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LogicalExpand

LogicalExpand[expr]

expands out logical combinations of equations, inequalities, and other functions.

Details and Options

  • LogicalExpand applies distributive laws for logical operations.
  • LogicalExpand generates ORs of ANDs corresponding to disjunctive normal form, with some contractions.

Examples

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Basic Examples  (1)

Expand out logic expressions:

Wolfram Language code: LogicalExpand[p && !(q || r)]

Scope  (7)

Represent logic expressions as a disjunction of conjunctions (disjunctive normal form):

Wolfram Language code: LogicalExpand[(a || b) && !(c || d || e)]

Simplify logic expressions:

Wolfram Language code: LogicalExpand[a || !a]
Wolfram Language code: LogicalExpand[a && !a]

Expand logic expressions out to forms involving only And, Or, and Not:

Wolfram Language code: LogicalExpand[Xor[p, q, r]]

Expand out combinations of equations:

Wolfram Language code: LogicalExpand[x == a && y == b || x == a || y == b]

Get separate equations for different terms in a series:

Wolfram Language code: Series[f[x] ^ 2, {x, 0, 2}] == 2 + x
Wolfram Language code: LogicalExpand[%]

Convert multi-argument equations and inequalities to two-argument form:

Wolfram Language code: x ≤ y + 1 < z && x ^ 2 - 1 == y ^ 2 - 2 == z ^ 2 - 3
Wolfram Language code: LogicalExpand[%]

Expand negations of Element statements:

Wolfram Language code: LogicalExpand[!Element[x | y | z, Reals]]

Applications  (1)

Prove tautologies:

Wolfram Language code: t = Implies[(p || !r) && s, Implies[r && s, p && q || p]]
Wolfram Language code: LogicalExpand[t]

Properties & Relations  (2)

The output of LogicalExpand is equivalent to the input:

Wolfram Language code: e1 = Implies[Xor[a, b, c], (a || b) && c]
Wolfram Language code: e2 = LogicalExpand[e1]
Wolfram Language code: Table[e1 == e2, {a, {True, False}}, {b, {True, False}}, {c, {True, False}}]

De Morgan's laws:

Wolfram Language code: LogicalExpand[!(a || b)]
Wolfram Language code: LogicalExpand[!(a && b)]

Neat Examples  (1)

An expanded Xor of variables will always contain terms:

Wolfram Language code: LogicalExpand[Xor@@Array[Subscript[x, #]&, 6]]//TraditionalForm
Wolfram Language code: Length[%]
Wolfram Research (1988), LogicalExpand, Wolfram Language function, https://reference.wolfram.com/language/ref/LogicalExpand.html.

Text

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

CMS

Wolfram Language. 1988. "LogicalExpand." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LogicalExpand.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_logicalexpand, author="Wolfram Research", title="{LogicalExpand}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/LogicalExpand.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_logicalexpand, organization={Wolfram Research}, title={LogicalExpand}, year={1988}, url={https://reference.wolfram.com/language/ref/LogicalExpand.html}, note=[Accessed: 12-June-2026]}