The Induction Principle

Abstract

The Induction Principle is of great importance in discrete mathematics: Number Theory, Graph Theory, Enumerative Combinatorics, Combinatorial Geometry, and other subjects. Usually one proves the validity of a relationship f(n) = g(n) if one has a guess from small values of n. Then one checks that f(n) = g(1), and, by making the assumption f(n) = g(n) for some n, one proves that also f(n + 1) = g(n + 1). From this one concludes by the Induction Principle that f(n) = g(n) for all n ɛ N.There are many variations of this principle. The relationship f(n) = g(n) is valid for 0 already, or, starting from some n₀ > 1. The inductive assumption is often f(k) = g(k) for all k < n, and, from this assumption, one proves the validity of f(n) = g(n). We assume familiarity with all this and apply induction in unusual circumstances to make nontrivial proofs. We refer to Polya [22] to [24] for excellent treatment of induction for beginners. The reader can acquire practice by proving some of the innumerable formulas for the Fibonacci sequence defined by F₀ = 0, F₁ = 1, F_n+2 = F_n+1 + F_n, n ≥ 0. We state some of these.