Theorems which state ``There exists an *x* such that *conclusion*'' are
known as existence theorems. Many, but not all,
theorems of this type are proved using a
*constructive proof* so named because one finds an element
*x* that does satisfy the conclusion.
As a simple example consider the statement ``Let *a* be
a real number. There exists a
real number *x* that is a solution to the equation 5*x*+3 = *a*''. A proof of
this statement would construct such a real number *x*. Using the rules of
arithmetic in the real number system, we know that if *a* is a real number
then so too is (*a*-3)/5. This is a value of *x* that is a solution
to the equation since 5*x* +3 = 5(*a*-3)/5 + 3 = *a* -3 +3 = *a* as required.
Of course not all proofs are this simple and to produce an element *x*
that will work may require ingenuity.

Sun Sep 15 22:27:27 PDT 1996