# 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, $(\text{Raj},5)$ is in the relation, but $(\text{Abdullah},7)$ 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 |

$\left(1,6\right)$ | $\left(3,-2\right)$ |

$\left(5,5.001\right)$ | $\left(-8,-9\right)$ |

$\left(0,9999\right)$ | $\left(4,3\right)$ |

## 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.