So, what is a 'relation'?
In math, a relation is just a set of ordered pairs.
Note:{ } are the symbol for "set"
Some Examples of Relations include- { (0,1) , (55,22), (3,-50) }
- { (0, 1) , (5, 2), (-3, 9) }
- { (-1,7) , (1, 7), (33, 7), (32, 7) }
One more time: A relation is just a set of ordered pairs. There is absolutely nothing special at all about the numbers that are in a relation. In other words, any bunch of numbers is a relation so long as these numbers come in pairs.
Video Lesson
What is the domain and range of a 'relation'?
Is the set of all the first numbers of the ordered pairs.
In other words, the domain is all of the x-values.
Is the set of the second numbers in each pair, or the y-values
Examples
of the Domain and Range
Example 2
Arrow Chart
Relations are often represented using arrow charts connecting the domain and range elements.

Example 3
Domain and range of a relation

In the relation above, the domain is { 2, 4, 11, -21}
the range is is { -5, 31, -11, 3}
I. Practice Identifying Domain and Range
Domain: -1, 2, 1, 8, 9
Range: 2, 51, 3, 22, 51
Domain: -5, 21, 11, 81, 19
Range: 6, -51, 93, 202, 51

Interactive Relation
See this demonstration by itself
What makes a relation a function?
Functions are a special kind of relation
At first glance, a function looks just like a relation.
- It's a set of ordered pairs such as { (0,1) , (5, 22), (11,9) }
- Like a relation, a function has a domain and range made up of the x and y values of ordered pairs.
In mathematics, what distinguishes a function from a relation is that each x value in a function has one and only ONE y-value.

Since relation #1 has ONLY ONE y value for each x value, this relation is a function
On the other hand, relation #2 has TWO distinct y values 'a' and 'c' for the same x value of '5' . Therefore, relation #2 does not satisfy the definition of a mathematical function
Teachers has multiple students
If we put teachers into the domain and students into the range, we do not have a function because the same teacher, like Mr. Gino below, has more than 1 student in a classroom.

Mothers and Daughters Analogy
A way to try to understand this concept is to think of how mothers and their daughters could be represented as a function.
Each element in the domain, each daughter , can only have 1 mother (element in the range).

Some people find it helpful to think of the domain and range as people in romantic relationships. If each number in the domain is a person and each number in the range is a different person, then a function is when all of the people in the domain have 1 and only 1 boyfriend/girlfriend in the range.
Compare the two relations on the below. They differ by just one number, but only one is a function.
What's an easy way to do this?

II. Practice Identifying Functions
- Relation #1 { (-1,2), (-4,51), (1,2), (8,-51) }
- Relation #2 { (13,14), (13,5), (16,7), (18,13) }
- Relation #3 { (3,90), (4,54), (6,71), (8,90) }
Relation #1 and Relation #3 are both functions.
- Relation #1{ (3,4), (4,5), (6,7), (8,9) }
- Relation #2 { (3,4), (4,5), (6,7), (3,9) }
- Relation #3{ (-3,4), (4,-5), (0,0), (8,9) }
- Relation #4 { (8, 11), (34,5), (6,17), (8,19) }
Relation #1 and Relation #3 are functions because each x value, each element in the domain, has one and only only one y value, or one and only number in the range.
- { (8 , 11), (34,5), (6,17), (X ,22) }
X cannot be 8, 34, or 6.
If x were 8 for instance, the relation would be:
{ (8, 11), (34,5), (6,17), (8 ,22) }
In this relation, the x-value of 8 has two distinct y values.
Therefore this relation would NOT be a function since each element in the domain must have 1 and only value in the range.
- { ( 12, 13), (-11, 22), (33, 101), (X ,22) }
X cannot be 12 or 33
If x were 12 for instance, the relation would be:
{ (12 , 13 ), (-11 , 22), ( 33, 101), (12 ,22 }
In this problem, x could be -11. Since (-11, 22) is already a pair in our relation, -11 can again go with a range element of 22 without creating a problem (We would just have two copies of 1 ordered pair).
If x were -11 , the relation would still be a function :
{ (12, 13 ), (-11 , 22) , ( 33, 101), (-11 ,22) }
The all important rule for a function in math -- that each value in the domain has only 1 value in the range-- would still be true if we had a second copy of 1 ordered pair.
- { (12,14), (13,5) , (-2,7), (X,13) }
X cannot be 12, 13, or -2.
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Further Reading
- Functions Home
- Vertical Line Test
- One to one function
- Inverse of a Function
- Images