Meaning of a Relation
Simple Explanation
A relation from set A to set B is any subset of the product set A × B — in other words, any chosen collection of ordered pairs linking elements of A to elements of B. The domain is the set of all first elements (from A) actually used; the range is the set of all second elements (from B) actually used.
Why Do We Need It?
Relations are the general category that functions belong to — understanding what makes a relation lets you later recognize exactly what extra condition makes a relation a function.
Worked Example
Find the domain and range of a relation
A relation is given by R = {(1,2), (2,4), (3,4), (4,6)}. Find its domain and range.
Why Does This Work?
A relation is defined purely by which ordered pairs it contains, so scanning every pair for its first and second entries — without duplicating repeated values, since sets don't repeat elements — directly gives the domain and range.
Real-Life Example
Student-to-club sign-up sheet
A sign-up sheet lists which students belong to which after-school clubs, with a student possibly joining more than one.
That whole sign-up list is a relation from the set of students to the set of clubs — one that is not necessarily a function, since one student can pair with several clubs.
Practice
What is the range of the relation {(2,5), (3,5), (4,7)}?
MediumCommon mistake
Listing the range with repeated values (e.g. {5, 5, 7}) — a set never lists the same value twice, even if it comes from two different pairs.
Quick Review
- A relation is any subset of a product set A × B — a chosen collection of ordered pairs.
- The domain is the set of all first coordinates used; the range is the set of all second coordinates used.
- Every function is a relation, but not every relation is a function.