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 "exactly one output per input" requirement is what distinguishes a function from an arbitrary correspondence. The image (or range) is the subset of $B$ actually produced as outputs; it may be smaller than the whole codomain.

Functions are classified by how they map:

• Injective (one-to-one) — different inputs give different outputs: $f(a) = f(a') \Rightarrow a = a'$. No two inputs collide.
• Surjective (onto) — every element of the codomain is an output: the image is all of $B$.
• Bijective — both injective and surjective, a perfect pairing between domain and codomain.

Bijections are exactly the invertible functions: a bijection $f$ has a unique inverse $f^{-1}$ satisfying $f^{-1}(f(a)) = a$. They also formalize "same size" — two sets have equal cardinality precisely when a bijection exists between them, a definition powerful enough to compare infinite sets (which is how we learn $\mathbb{Q}$ is countable but $\mathbb{R}$ is not).

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 $g$ — read right to left. Composition is associative, and the composition of two bijections is again a bijection. Functions are the connective tissue of mathematics, linking one structure to another while preserving or transforming information.

Common pitfall: confusing injective (one-to-one) with surjective (onto), and reading composition left to right. Injective forbids input collisions; surjective requires the codomain to be fully covered — a function can be one without the other. And $g \circ f$ means "$f$ then $g$": the function on the right acts first. Reversing the order gives the wrong result whenever $f$ and $g$ do not commute.

A domain column and codomain column of dots joined by arrows; an accent collision (two arrows into one target) fails injectivity, and an unhit target fails surjectivity.

$$f(a) = f(a') \Rightarrow a = a'$$