Functions: partial, total

A function is a special kind of relation: $f \subseteq X \times Y$ where every
$x \in X$ is paired with AT MOST one $y \in Y$ (right-unique).

• Partial function: $f$ defined on a subset of $X$.
• Total function: $f$ defined on ALL of $X$ (for every $x \in X$, exactly one $y$).
• Domain $\text{dom}(f)$: the set of inputs where $f$ is defined.
• Codomain $\text{codom}(f)$: the set $Y$ (the "type" of the output).
• Range / image $f(X)$: the actual outputs produced.

In SML: the expression fn x => x + 1 is a function. The expression 1 / 0 is undefined (partial).