# Definition:Domain (Relation Theory)/Mapping

## Definition

Let $f: S \to T$ be a mapping.

The domain of $f$ is $S$, and can be denoted $\Dom f$.

## Also known as

The domain of (usually) a mapping is sometimes called the departure set.

Some sources refer to $\Dom f$ as the domain of definition of $f$.

Others refer to it on occasion as the source, but this is not recommended as there are other uses for that term.

1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability, for example, possibly forgetting themselves, in Appendix $\text{A}.7$:

Here are some common functions and their inverses. Note how carefully the source and codomain are specified.

Some sources denote the domain of $f$ by $\map {\mathrm D} f$.

Some sources use $D_f$.

## Examples

### Arbitrary Example

Let $f$ be defined as:

$\forall x: 0 \le x \le 2: \map f x = x^3$

The domain of $f$ is the closed interval $\closedint 0 2$.

## Also see

• Results about domains of mappings can be found here.

## Technical Note

The $\LaTeX$ code for $\Dom {X}$ is \Dom {X} .

When the argument is a single character, it is usual to omit the braces:

\Dom X