Two Important Principles: Induction and Pigeonhole
Mathematical propositions come in two forms: universal propositions which state that something is true for all values of x in some specified set, and existential propositions which state that something is true for some value of x in some specified set. The former type are expressible in the form “For all x (in a set S), P(x)”; the latter type are expressible in the form “There exists an x (in the set S) such that P(x),” where P(x) is a statement about x. In this chapter we will consider two important techniques for dealing with these two kinds of statements: (i) the principle of mathematical induction, for universal propositions, and (ii) the pigeonhole principle, for existential propositions.
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