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 x-values 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 x-values (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 x-values on this line have one (and only one) y-value 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 x-value has only one y-value. 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 y-value associated with the x-values. 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 x-values for which there is more than one associated y-value. ; 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 x-value associated with only ONE y-value, or it may have some (or all) x-values associated with more than ONE y-value. " Relations are willing to choose one or more partners."