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Search
This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)
Mathematica
>
Mathematics and Algorithms
>
Formula Manipulation
>
Assumptions and Domains
>
ForAll (
)
>
BUILTIN MATHEMATICA SYMBOL
Quantifiers
Tutorials »

Exists
Resolve
Conjunction
Reduce
Element
Blank
SolveAlways
TautologyQ
See Also »

Assumptions and Domains
Boolean Computation
Formula Manipulation
Logic & Boolean Algebra
Polynomial Systems
More About »
ForAll
ForAll
represents the statement that
expr
is
True
for all values of
.
ForAll
states that
expr
is
True
for all
x
satisfying the condition
cond
.
ForAll
states that
expr
is
True
for all values of all the
.
MORE INFORMATION
ForAll
can be entered as
. The character
can be entered as
Esc
fa
Esc
or
\[ForAll]
. The variable
is given as a subscript.
ForAll
can be entered as
.
In
StandardForm
,
ForAll
is output as
.
ForAll
is output as
.
ForAll
can be used in such functions as
Reduce
,
Resolve
, and
FullSimplify
.
The condition
cond
is often used to specify the domain of a variable, as in
x
Integers
.
ForAll
is equivalent to
ForAll
[
x
,
Implies
[
cond
,
expr
]]
.
ForAll
is equivalent to
.
The value of
in
ForAll
is taken to be localized, as in
Block
.
EXAMPLES
CLOSE ALL
Basic Examples
(1)
This states that for all
,
is positive:
Use
Resolve
to get a condition on real parameters for which the statement is true:
Reduce
gives the condition in a solved form:
This states that for all
,
is positive:
In[1]:=
Out[1]=
Use
Resolve
to get a condition on real parameters for which the statement is true:
In[2]:=
Out[2]=
Reduce
gives the condition in a solved form:
In[3]:=
Out[3]=
Scope
(6)
This states that for all
the inequation is true:
Use
Resolve
to prove that the statement is false:
This states that for all real
the inequation is true:
Use
Resolve
to prove that the statement is true:
This states that for all pairs
the inequality is true:
With domain not specified,
Resolve
considers algebraic variables in inequalities to be real:
With domain
Complexes
, complex values that make the inequality
False
are allowed:
This states the tautology
implies
:
Prove it:
If the expression does not explicitly contain a variable,
ForAll
simplifies automatically:
TraditionalForm
formatting:
Applications
(5)
This states the inequality between the arithmetic mean and the geometric mean:
Use
Resolve
to prove the inequality:
This states a special case of Hölder's inequality:
Use
Resolve
to prove the inequality:
This states a special case of Minkowski's inequality:
Use
Resolve
to prove the inequality:
Prove geometric inequalities for
,
, and
sides of a triangle:
This states that an inequality is satisfied for all triangles:
Use
Resolve
to prove the inequality:
This states that an inequality is satisfied for all acute triangles:
Use
Resolve
to prove the inequality:
Test whether one region is included in another:
This states that all points satisfying
also satisfy
:
The statement is true, hence the region defined by
is included in the region defined by
:
Plot the relationship:
Properties & Relations
(3)
Negation of
ForAll
gives
Exists
:
Quantifiers can be eliminated using
Resolve
or
Reduce
:
This eliminates the quantifier:
This eliminates the quantifier and solves the resulting equations and inequalities:
This states that an equation is true for all complex values of
:
Use
Reduce
to find the values of parameters for which the statement is true:
This solves the same problem using
SolveAlways
:
SEE ALSO
Exists
Resolve
Conjunction
Reduce
Element
Blank
SolveAlways
TautologyQ
TUTORIALS
Quantifiers
MORE ABOUT
Assumptions and Domains
Boolean Computation
Formula Manipulation
Logic & Boolean Algebra
Polynomial Systems
RELATED LINKS
NKSOnline
(
A New Kind of Science
)
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