

A relation is simply a set of input and output values, represented in ordered pairs.
It is a relationship between sets of information. 

A relation can be any set of ordered pairs.
No special rules need apply to a "relation".
The following is an example of a relation:
{(1,1),(1,2),(3,3),(4,4),(5,5),(5,6),(6,4)}
NOTICE: In a relation, points can be plotted one above the other on a graph. The ordered pairs can have the xvalues repeated, such as (1,1) and (1,2). The red vertical dashed lines on the graph
show
where this happens.

This graph is a "relation":
{(1,1),(1,2),(3,3),(4,4),(5,5),(5,6),(6,4)}

As seen above, a relation can be expressed in a graph,
and can be expressed in set notation: {(1,1),(1,2),(3,3),(4,4),(5,5),(5,6),(6,4)}
Relations can also be expressed
in a table:
x 
y 
1 
1 
1 
2 
3 
3 
4 
4 
5 
5 
5 
6 
6 
4 

Relations can also be expressed
in a mapping diagram:


Consider this example of a relation:
The relationship between eye color and student names.
(x,y) = (eye color, student's name)
Set A = {(green, Steve), (blue, Elaine), (brown, Kyle), (green, Marsha), (blue, Miranda), (brown, Dylan)}
Notice that the xvalues (eye colors) get repeated. 
The graph we saw at the top of this page was a "scatter plot" which is comprised of a series of individual points, not connected.
A relation can also be a "connected" graph such as the graph shown at the right (a straight line).
This is the graph of y = x. Unlike the scatter plot, the xvalues on this line have one (and only one) yvalue associated with each of them.
If a vertical line is drawn on this graph, the line would only intersect the graph in ONE location, showing each xvalue has only one yvalue.

This graph is a "relation".
We will see in upcoming lessons that it is
a "special" type of relation (called a function).

It is also possible that a "connected graph" can have more than one yvalue associated with the xvalues.
The graph at the right is the graph of the square root of x, assuming only values of 0 or larger are used for x.
The red vertical dashed line on the graph shows that there are xvalues for which there is more than one associated yvalue.
; allows for points
such
as
(4, 2) and
(4,2), or (2,1.424) and (2,1.414) to exist.

This graph is a "relation".


The thing to remember about "relations" and graphs:
... a relation may have every xvalue associated with only ONE yvalue,
or
it may have some (or all) xvalues associated with more than ONE yvalue.
"
Relations are willing to choose one or more partners."

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