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Exists (Exists)

Exists[x, expr]
represents the statement that there exists a value of x for which expr is True.
Exists[x, cond, expr]
states that there exists an x satisfying the condition cond for which expr is True.
Exists[{x1, x2, ...}, expr]
states that there exist values for all the x_i for which expr is True.
  • Exists[x, expr] can be entered as exists _xexpr. The character exists can be entered as Esc ex Esc or \[Exists]. The variable x is given as a subscript.
  • Exists[x, cond, expr] can be entered as exists _(x,cond)expr.
  • Exists[x, cond, expr] is output as exists _(x,cond)expr.
  • The condition cond is often used to specify the domain of a variable, as in xElementIntegers.
  • Exists[x, cond, expr] is equivalent to Exists[x, cond&&expr].
  • Exists[{x1, x2, ...}, ...] is equivalent to exists _(x_1) exists _(x_2)....
  • The value of x in Exists[x, expr] is taken to be localized, as in Block.
EXAMPLESCLOSE ALL
Basic Examples   (1)
This states that there exists a positive solution to the equation a x^2+b x+c=0:
In[1]:=
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Out[1]=
Use Resolve to get a condition on real parameters for which the statement is true:
In[2]:=
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Out[2]=
Reduce gives the condition in a solved form:
In[3]:=
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Out[3]=
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