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Conjunction[expr,{a1,a2,…}]

gives the conjunction of expr over all choices of the Boolean variables ai.

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Conjunction

Conjunction[expr,{a1,a2,…}]

gives the conjunction of expr over all choices of the Boolean variables ai.

Details

Examples

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Basic Examples  (3)

The conjunction over a set of variables:

Wolfram Language code: Conjunction[f[a, b], {a, b}]

Show that a formula is a tautology:

Wolfram Language code: Conjunction[Equivalent[a∧(b∨c), a∧b∨a∧c], {a, b, c}]

Find the conditions on a for ab to be true for any b:

Wolfram Language code: Conjunction[Implies[a, b], {b}]

Properties & Relations  (5)

Conjunction effectively computes the And over all truth values of the listed variables:

Wolfram Language code: And@@BooleanTable[a && b && (c || d) || Xor[a, c, d], {c, d}]
Wolfram Language code: Conjunction[a && b && (c || d) || Xor[a, c, d], {c, d}]
Wolfram Language code: TautologyQ[Equivalent[%%, %]]

Conjunction is typically more efficient and can handle large numbers of variables:

Wolfram Language code: f = BooleanConvert[BooleanCountingFunction[{30, 70}, 100], "BFF"]
Wolfram Language code: Conjunction[f@@Array[x, 100], Array[x, 40]]

Conjunction effectively eliminates ∀ (ForAll) quantifiers for the list of variables:

Wolfram Language code: Resolve[ForAll[{c, d}, a && b && (c || d) || Xor[a, c, d]]]
Wolfram Language code: Conjunction[a && b && (c || d) || Xor[a, c, d], {c, d}]

Use Resolve to eliminate more general combinations of quantifiers:

Wolfram Language code: Resolve[Subscript[∀, a]Subscript[∃, b]Xor[a, b, c]]

TautologyQ is Conjunction over all variables:

Wolfram Language code: TautologyQ[Implies[Implies[a, b]∧a, b]]
Wolfram Language code: Conjunction[Implies[Implies[a, b]∧a, b], {a, b}]

Use Disjunction to compute Or over a list of variables:

Wolfram Language code: Disjunction[f[a, b, c], {b, c}]

Disjunction is related to Conjunction by de Morgan's law:

Wolfram Language code: TautologyQ[Equivalent[¬Conjunction[¬f[a, b], {a, b}], Disjunction[f[a, b], {a, b}]]]

Conjunction is repeated And, just as Product is repeated Times:

Wolfram Language code: Product[f[i, j], {i, 0, 1}, {j, 0, 1}]
Wolfram Language code: Conjunction[f[i, j], {i, j}]

Represent Conjunction in terms of Product:

Wolfram Language code: And@@Product[f[i, j], {i, {True, False}}, {j, {True, False}}]
Wolfram Research (2008), Conjunction, Wolfram Language function, https://reference.wolfram.com/language/ref/Conjunction.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_conjunction, author="Wolfram Research", title="{Conjunction}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Conjunction.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_conjunction, organization={Wolfram Research}, title={Conjunction}, year={2008}, url={https://reference.wolfram.com/language/ref/Conjunction.html}, note=[Accessed: 12-June-2026]}