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Existence Proofs

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 5x+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 5x +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.

Peter Williams
Sun Sep 15 22:27:27 PDT 1996