BUILT-IN MATHEMATICA SYMBOL

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 xIntegers.

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: