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Implies[p,q]

represents the logical implication .

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Implies

Implies[p,q]

represents the logical implication .

Details

Examples

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Basic Examples  (2)

Enter using =>:

Scope  (4)

Certain arguments automatically simplify:

Expand in terms of And, Or, and Not:

Do symbolic simplification:

TraditionalForm formatting:

Applications  (3)

Use Implies to combine two regions:

Symbolic reduction:

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

This shows the set:

Use Implies to express set inclusion:

Properties & Relations  (2)

Truth table for binary Implies:

Implication of conditions in Boole functions:

Wolfram Research (1988), Implies, Wolfram Language function, https://reference.wolfram.com/language/ref/Implies.html (updated 1996).

Text

Wolfram Research (1988), Implies, Wolfram Language function, https://reference.wolfram.com/language/ref/Implies.html (updated 1996).

CMS

Wolfram Language. 1988. "Implies." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/Implies.html.

APA

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

BibTeX

@misc{reference.wolfram_2025_implies, author="Wolfram Research", title="{Implies}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Implies.html}", note=[Accessed: 28-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_implies, organization={Wolfram Research}, title={Implies}, year={1996}, url={https://reference.wolfram.com/language/ref/Implies.html}, note=[Accessed: 28-August-2025]}