Mathematical Logic, Induction, Proofs

Chapter Summary

This chapter began with some cautionary remarks pertaining to conducting formal proofs: how one has to spend some time assessing whether the proof goal is achievable, and then how to carry out the proof in a manner that is easily verifiable. It then discussed the operators ‘if’ (⇒) and ‘iff’ (⇔). Proof by contradiction was introduced through the game of Mastermind. After discussing quantifications, inductively defined sets and functions, and induction principles, a proof of equivalence between arithmetic and complete induction was given. Various other induction principles were also discussed. Many of these concepts were illustrated by the pigeon-hole principle.