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'?

The domain:

is the set of all the first numbers of the ordered pairs .
In other words, the domain is all of the x-values.

The range:

is the set of the second numbers in each pair, or the y-values

Answer 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.

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?

Look for repeated elements in the domain

As soon as an element in the domain repeats, watch out!

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.

Practice 3) For the following relation to be a function, X can not be what values?

{ (8 , 11), (34,5), (6,17), (X ,22) }

Answer

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.

Practice 4) For the relation below to be a function, X cannot be what values?

{ ( 12, 13), (-11, 22), (33, 101), (X ,22) }

Answer

X cannot be 12 or 33 .

If x were 12 for instance, the relation would be:

{ (12 , 13 ), (-11 , 22), ( 33, 101), (12 ,22 }

Did we trick you?

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 :

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.

Problem 5)
For the relation below to be a function, X cannot be what values?