Existential Proofs

Abstract

In the last several chapters we discussed proof techniques for universal statements of the form

$$\displaystyle{\forall x \in U,P(x);}$$

in this chapter we focus on the existential quantifier and analyze existential statements of the form

$$\displaystyle{\exists x \in U,P(x).}$$

For instance, we may claim that a certain equation has a real number solution (the existence of $\sqrt{2}$, to be formally proven only in Chap. 23, is a prime example), or we may claim that a certain set has a minimum element (by Theorem 13.6, every nonempty set of natural numbers does). Quite often, we deal with statements of the form

$$\displaystyle{\forall x \in U,\exists y \in V,P(x,y);}$$

for example, when in Chap. 1 we claimed that a certain game had a winning strategy for Player 2, we made an existential statement that for any sequence of moves by Player 1, there was a response by Player 2 that resulted in a win for Player 2.