# Question af1dd

Apr 5, 2017

See below for details.

#### Explanation:

When relations are defined by a set of paired values,
the first value in each pair is a Domain element
and the second value is a Range element.
We typically say that, for each pair the Domain element "maps into" the Range element.

A relation is a function if no Domain value "maps into" more than one Range value.

For the relations:
A= {(1, 2); (2, 3); (3, 4); (2, 5)}

the pairs $\left(2 , 3\right)$ and $\left(2 , 5\right)$ map the same Domain value ($2$) into different Domain values ($3$ and $5$);
therefore $A$ is not a function.

B= {(1, 2); (1, 3); (3, 2); (4, 2)}

the pairs $\left(1 , 2\right)$ and $\left(1 , 3\right)$ map the Domain value ($1$) into different Range values;
$B$ is not a function.

C: {(1, 2); (2, 3); (3, 4); (1, 5)}

the pairs $\left(1 , 2\right)$ and $\left(1 , 5\right)$ map the Domain value ($1$) into different Range values;
$C$ is not a function.

D= {(1, 2); (2, 5); (3, 2); (4, 5)} #

there are no Domain values which map into more than one Range value;
$D$ is a function.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The Range of the relation $\left\{\begin{matrix}1 & 2 \\ 2 & 4 \\ 3 & 2\end{matrix}\right\}$
is the set of Range values, namely $\left\{2 , 4\right\}$ (it is not necessary to include the value $2$ in the Range set more than once).