Problem-Solving Through Problems pp 58-83 | Cite as

# Two Important Principles: Induction and Pigeonhole

## Abstract

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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