Math101 · Independent Ontario learning library
Math101Relations
A relation is any set of ordered pairs connecting inputs with outputs. The domain is the set of first coordinates and the range is the set of second coordinates.
Relations provide the general language for pairing quantities, while the function condition underlies algebraic models. Domain and range keep input and output restrictions explicit.
Intuition and core definition
A relation is any set of ordered pairs connecting inputs with outputs. The domain is the set of first coordinates and the range is the set of second coordinates. A function is a special relation in which each input is paired with exactly one output; different inputs may share an output.
Notation, language, and conditions
A relation can be written as a set, table, mapping diagram, graph, or equation. Repeated elements are listed once in a domain or range set. The vertical line test detects whether a graph represents $y$ as a function of $x$: any vertical line may intersect at most once.
Why this idea matters
A relation pairs inputs with outputs, while a function adds the restriction that each input may select only one output.
A dependable method
- List or read every ordered pair.
- Collect unique first coordinates to form the domain.
- Collect unique second coordinates to form the range.
- Check whether any one input is paired with two different outputs.
- Confirm the classification in a second representation when possible.
