Relations

A relation is simply a set of ordered pairs . Usually, we talk about relations on sets of numbers, but not always.

Example 1:

You could have a relation between the set of all names and the set of whole numbers. A name $N$ is related to a number $x$ if and only if $N$ has fewer than $x$ letters.

So, $(Raj, 5)$ is in the relation, but $(Abdullah, 7)$ is not.

Example 2:

Here is a relation on the set of real numbers. Suppose $x$ is related to $y$ if and only if $x$ is less than $y$ .

The following table shows some ordered pairs which are in the relation, and some which are not.

Related	Not Related
$(1, 6)$	$(3, - 2)$
$(5, 5.001)$	$(- 8, - 9)$
$(0, 9999)$	$(4, 3)$

Input-Output Tables

One way in which relations are commonly displayed is in an input-output table. The idea is, you input some number $x$ and you get out some $y$ .

Input	Output
$0$	$0$
$1$	$3$
$2$	$0$
$3$	$9$
$1$	$3$
$- 5$	$- 15$

This table describes a relation containing the ordered pairs $(0, 0), (1, 3), (2, 0), (3, 9), (1, 3), (- 5, - 15)$ .

If the same input always gives the same output, then the relation is called a function . Otherwise it is not a function. The relation in the table above is a function (it is okay if two different inputs give the same output). The relation in the table below is not a function because the same input $1$ gives the output $5$ the first time and $0$ the second time.

Input	Output
$0$	$0$
$1$	$5$
$2$	$0$
$3$	$15$
$1$	$0$
$- 5$	$- 15$

If you have a graph of a relation, you can use the vertical line test to decide whether or not it is a function.

Relations

Input-Output Tables

Download our free learning tools apps and test prep books