Recurrence Relations

Abstract

Combinatorics is the study of finite sets. To define finite sets, we need the notion of bijective function. Given two sets X and Y, a function f : X → Y is injective or one-to-one if f(a) ≠ f(b) for any a, b ∈ X with a ≠ b. A function f : X → Y is surjective or onto if for any y ∈ Y, there exist x ∈ X such that f(x) = y. A function is bijective if it is injective and surjective. A function f : X → Y is invertible if there exists a function g : Y → X such that f(x) = y if and only if g(y) = x. If g exists, it is called the inverse of f and it is usually denoted by f⁻¹. We leave as an exercise the fact that a function is bijective if and only if it is invertible.