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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 x_i.
  • ForAll[x, expr] can be entered as ∀_xexpr. The character ∀ can be entered as Esc fa Esc or \[ForAll]. The variable x is given as a subscript.
  • ForAll[x, cond, expr] can be entered as ∀_(x,cond)expr.
  • ForAll[x, cond, expr] is output as x, condexpr.
  • The condition cond is often used to specify the domain of a variable, as in xIntegers.
  • ForAll[{x1, x2, ...}, ...] is equivalent to ∀_(x_1)(∀_(x_2)…).
  • The value of x in ForAll[x, expr] is taken to be localized, as in Block.
This states that for all x, a x^2+b x+c 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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