Implies
Implies[p,q]
represents the logical implication 
.
Details
- As a Boolean function, Implies[p,q] is equivalent to
. - Implies[p,q] can be input in StandardForm and InputForm as
. The character can be entered as 
=>
or \[Implies].
Examples
open all close allBasic Examples (3)
Typeset output automatically uses the corresponding character:
Implies[True, a]Implies[False, a]
:pqAn input using the typeset character gives an Implies expression:
pq//FullFormScope (4)
Certain arguments automatically simplify:
{Implies[False, x], Implies[True, x]}Expand in terms of And, Or, and Not:
BooleanConvert[Implies[x, y]]Implies[p, Implies[q, p]]//SimplifyTraditionalForm formatting:
Implies[x, y]//TraditionalFormApplications (3)
Use Implies to combine two regions:
RegionPlot[Implies[x ^ 2 + y ^ 2 < 1, x + y > 0], {x, -2, 2}, {y, -2, 2}]Reduce[Implies[x ^ 2 + y ^ 2 < 1, x + y > 0], {x, y}, Reals]Find the area of the complement of the difference of sets given by algebraic conditions:
Integrate[Boole[Implies[x ^ 2 + y ^ 2 < 1, (x - 1) ^ 2 + y ^ 2 < 2]], {x, -3 / 2, 5 / 2}, {y, -2, 2}]//FullSimplifyRegionPlot[Implies[x ^ 2 + y ^ 2 < 1, (x - 1) ^ 2 + y ^ 2 < 2], {x, -3 / 2, 5 / 2}, {y, -2, 2}]Use Implies to express set inclusion:
Resolve[ForAll[{x, y}, Implies[x ^ 2 + 2y ^ 2 < 1, x ^ 2 + y ^ 2 < 1]]]RegionPlot[{x ^ 2 + y ^ 2 < 1, x ^ 2 + 2y ^ 2 < 1}, {x, -1, 1}, {y, -1, 1}]Related Links
NKS|OnlineText
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