Geometric Bites

A Gentle Introduction to Function Notation Understanding f : A → B — the language of modern mathematics. One of the most powerful ideas in mathematics is the concept of a function . We usually meet it in the form f(x) = 2x + 1 , but the structure behind this idea is far richer. The notation f : A → B captures the entire architecture of a function in a single line. In this article, we unpack this notation and explain exactly what it means, why it matters, and how it connects to the familiar expression f(x) = y . 1. What does f : A → B mean? When we write f : A → B we are saying: f is a function, A is the domain — the set of inputs the function accepts, B is the codomain — the set in which all outputs must lie. In words: A function assigns to every element of the domain A exactly one output in the codomain B . Two rules always hold for genuine functions: Every input must have an output. No input may have more than one output. D...

What Is an Isomorphism? In mathematics, an isomorphism is a function that shows two mathematical objects have the same structure. Although the objects may look different, an isomorphism demonstrates that they behave in exactly the same way with respect to the operations that define them. If such a map exists, the objects are called isomorphic . An isomorphism tells us that two systems are essentially the same, differing only by a relabelling of their elements. The Basic Idea An isomorphism is a function: f : A → B that must be: Injective — different elements of A map to different elements of B. Surjective — every element of B comes from some element of A. Together these mean f is bijective , and so it has an inverse: f⁻¹ : B → A No information is lost moving from A to B or back. Preserving Structure Bijectivity alone is not enough. An isomorphism must also preserve structure. For groups, this means: f(a ⋆ b) = f(a) ∘ f(b) for all a, b ∈ A ,...