Functions

Mapping Rules

A function $f: A \to B$ assigns to each element of the domain $A$ exactly one element of the codomain $B$. The image (range) is the set of outputs actually hit.

Three core properties describe how functions map sets:

Bijections are invertible, possessing an inverse $f^{-1}$ that undoes the mapping. They define set cardinality, letting us compare even infinite sets by checking if a bijection exists between them.

Composition and Pitfalls

Functions compose: given $f: A \to B$ and $g: B \to C$, the composition $g \circ f: A \to C$ applies $f$ first, then $g$. Programs are built from composed functions transforming data step-by-step.

Common Pitfall: Mixing up injective and surjective, or misreading composition order. Injective means inputs do not collide; surjective means the codomain is fully covered. Always remember that $g \circ f$ means apply $f$ then $g$—the right-hand function runs first, which looks backwards until you practice it.