Math101 · Independent Ontario learning library
Math101learn.math101.caRelations
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.
Worked example
Representations and interpretation
In a mapping diagram, arrows leaving one domain element expose the function condition. A graph encodes pairs as points; a table makes duplicates easy to scan. Each display represents the same set of pairings.
Reasoning about variations
The inverse relation swaps every coordinate. A function’s inverse relation may fail to be a function if the original gives the same output to multiple inputs. Thus many-to-one behaviour is allowed forward but creates one-to-many behaviour after swapping.
Common mistakes
How to check your work
- Trace every input and count its distinct outputs.
- Convert the table or mapping to ordered pairs and compare.
- On a graph, imagine vertical lines across the entire displayed domain.
Practice
- Is $\{(1,4),(2,4),(3,5)\}$ a function?
- Find the domain of $\{(-1,2),(0,2),(5,9)\}$.
- Does the circle $x^2+y^2=1$ define $y$ as a function of $x$?
Answers and brief solutions
Show answers
- Yes Each input appears with exactly one output; repeated output $4$ is allowed.
- $\{-1,0,5\}$ The domain contains the unique first coordinates.
- No Most vertical lines through the circle meet it twice.
Synthesis and transfer
A student-to-course enrollment relation can assign several students to one course, but it ceases to be a function from students if one student is paired with two course outputs.
A relation can be displayed as ordered pairs, a mapping diagram, a table, or a graph, and the function test must agree in every representation. Reversing all pairs forms the inverse relation; even if the original is a function, its inverse may assign one input to several outputs. In the enrollment example, course-to-student is usually one-to-many and therefore not a function if each course input is expected to have one student output. Restricting the domain or codomain can change that classification without altering the stored pairs. The vertical-line test is a graphical version of the same input-uniqueness condition, not an unrelated shortcut.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Is $\{(1,4),(2,4),(3,5)\}$ a function?
- Each input appears with exactly one output; repeated output $4$ is allowed.
End of lesson
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