Domain

Let $f : A \to B$ be a function.

Definition. The domain of $f$ is the set $A$. We write $\mathrm{dom}(f) = A$.

Every element $a \in A$ has exactly one value $f(a) \in B$. The function $f$ is undefined outside $\mathrm{dom}(f)$.

Proposition. Two functions $f, g : A \to B$ with the same domain, codomain, and rule ($f(a) = g(a)$ for all $a \in A$) are equal. Two functions with the same rule but different domains are distinct.

Definition. The codomain of $f : A \to B$ is the set $B$. The image (or range) of $f$ is $\mathrm{im}(f) = \{f(a) \mid a \in A\} \subseteq B$.

Proposition. $\mathrm{im}(f) \subseteq \mathrm{cod}(f)$, with equality iff $f$ is surjective.

In set theory, a function $f : A \to B$ is a subset $f \subseteq A \times B$ satisfying the unique-value condition: for each $a \in A$, there is exactly one $b \in B$ with $(a, b) \in f$.

In category theory, the domain (also called source) of a morphism $f : A \to B$ is the object $A$.

Open questions