ForAll

ForAll[x,expr]

represents the statement that expr is True for all values of x.

ForAll[x,cond,expr]

states that expr is True for all x satisfying the condition cond.

ForAll[{x1,x2,},expr]

states that expr is True for all values of all the xi.

Details

  • ForAll[x,expr] can be entered as xexpr. The character can be entered as fa or \[ForAll]. The variable x is given as a subscript.
  • ForAll[x,cond,expr] can be entered as x,condexpr.
  • In StandardForm, ForAll[x,expr] is output as xexpr.
  • ForAll[x,cond,expr] is output as x,condexpr.
  • 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[x,cond,expr] is equivalent to ForAll[x,Implies[cond,expr]].
  • ForAll[{x1,x2,},] is equivalent to .
  • The value of in ForAll[x,expr] is taken to be localized, as in Block.

Examples

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Basic Examples  (1)

This states that for all , is positive:

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Use Resolve to get a condition on real parameters for which the statement is true:

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Reduce gives the condition in a solved form:

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Scope  (6)

Applications  (5)

Properties & Relations  (3)

See Also

Exists  Resolve  Conjunction  Reduce  Element  Blank  SolveAlways  TautologyQ  AllTrue

Tutorials

Introduced in 2003
(5.0)