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Equivalent[e1,e2,…]

represents the logical equivalence e1⇔e2⇔…, giving True when all of the ei are the same.

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Equivalent

Equivalent[e1,e2,…]

represents the logical equivalence e1⇔e2⇔…, giving True when all of the ei are the same.

Details

Examples

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Basic Examples  (2)

Test equivalence of Boolean expressions:

Wolfram Language code: Equivalent[a && (b || c), a && b || a && c]//TautologyQ

Enter using equiv:

Wolfram Language code: a⧦b

Scope  (2)

Automatic simplifications:

Wolfram Language code: Equivalent[a, b, a, b, c]
Wolfram Language code: Equivalent[a, b, c, True]
Wolfram Language code: Equivalent[a, b, c, False]
Wolfram Language code: Equivalent[a, b, True, False]

TraditionalForm formatting:

Wolfram Language code: Equivalent[a, b, c]//TraditionalForm

Applications  (1)

Prove equivalence between different Boolean expressions:

Wolfram Language code: f = BooleanFunction[33, {a, b, c}, "DNF"]
Wolfram Language code: g = BooleanFunction[33, {a, b, c}, "CNF"]
Wolfram Language code: TautologyQ[Equivalent[f, g]]

Properties & Relations  (7)

Truth table for binary Equivalent:

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

Ternary Equivalent:

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

Use BooleanConvert to express Equivalent in terms of And and Or:

Wolfram Language code: BooleanConvert[Equivalent[a, b, c]]

A well-known representation of two-argument Equivalent in terms of Implies:

Wolfram Language code: e1 = Equivalent[a, b]; e2 = Implies[a, b] && Implies[b, a];

This proves that the two representations are indeed equivalent:

Wolfram Language code: TautologyQ[Equivalent[e1, e2]]

Equivalent can be represented in terms of BooleanCountingFunction:

Wolfram Language code: equivalent[v__] := BooleanCountingFunction[{{0, Length[{v}]}}, Length[{v}]][v]
Wolfram Language code: TautologyQ[Equivalent[Equivalent@@Array[a, 5], equivalent@@Array[a, 5]]]

Equivalent with two arguments is equivalent to Xnor:

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

For more arguments, these are different primitives:

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

Use Resolve to prove equivalence of two systems of equations:

Wolfram Language code: ForAll[{x, y}, Equivalent[x y == 1 && x ^ 2 + y ^ 2 == 2, x == y && x ^ 2 == 1]]
Wolfram Language code: Resolve[%]

Equivalent is effectively Equal for Boolean expressions:

Wolfram Language code: BooleanTable[a == b == c, {a, b, c}]
Wolfram Language code: BooleanTable[Equivalent[a, b, c], {a, b, c}]
Wolfram Language code: %% == %
Wolfram Research (2008), Equivalent, Wolfram Language function, https://reference.wolfram.com/language/ref/Equivalent.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_equivalent, author="Wolfram Research", title="{Equivalent}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Equivalent.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_equivalent, organization={Wolfram Research}, title={Equivalent}, year={2008}, url={https://reference.wolfram.com/language/ref/Equivalent.html}, note=[Accessed: 13-June-2026]}