Functions as Mappings

Rules That Assign

A function $f : A \to B$ is a rule assigning to each element of the domain $A$ exactly one element of the codomain $B$. The image (range) is the subset of $B$ actually produced; it may be smaller than the whole codomain.

Functions are classified by how they map:

Bijections are invertible: a bijection $f$ has a unique inverse $f^{-1}$ satisfying $f^{-1}(f(a)) = a$. They formalize equal cardinality, allowing us to compare even infinite sets.

Composition & Pitfalls

Functions compose: for $f : A \to B$ and $g : B \to C$, the composition $(g \circ f)(x) = g(f(x))$ applies $f$ first, then $g$. Composition is associative, and the composition of two bijections is again a bijection.

$$\text{Order matters: } (g \circ f)(x) = g(f(x))$$

Common pitfall: Confusing injective with surjective, and reading composition left to right. Injective forbids input collisions; surjective requires full codomain coverage. In $g \circ f$, the function on the right ($f$) acts first. Reversing this order yields the wrong result.