# How do you find the domain and the range of the relation, and state whether or not the relation is a function (1,0), (2,0), (3,0), (4,0)?

Feb 19, 2018

$\text{1) domain of} \setminus \setminus R \setminus = \setminus \setminus \left\{1 , 2 , 3 , 4 \setminus\right\} .$

$\text{2) range of} \setminus \setminus R \setminus = \setminus \setminus \left\{1 , 2 , 3 , 4 \setminus\right\} .$

$\text{3)" \ \ \ R \ \ "is a function.}$

#### Explanation:

$\text{We are given a relation, call it" \ R:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad R \setminus = \setminus \setminus \left\{\begin{matrix}1 & 0 \\ 2 & 0 \\ 3 & 0 \\ 4 & 0 \setminus\end{matrix}\right\} .$

$\text{1) Recall that the domain of a relation is the set of all first}$
$\text{coordinates of the ordered pairs in the relation. So:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \text{domain of} \setminus \setminus R \setminus = \setminus \setminus \left\{1 , 2 , 3 , 4 \setminus\right\} .$

$\text{2) Recall that the range of a relation is the set of all second}$
$\text{coordinates of the ordered pairs in the relation. So:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \text{range of} \setminus \setminus R \setminus = \setminus \setminus \left\{1 , 2 , 3 , 4 \setminus\right\}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus = \setminus \setminus \left\{0 \setminus\right\} .$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \text{(remember to simplify the set;}$
$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \text{duplicate entries must be removed)}$

$\text{3) Recall that a relation is a function precisely when the first}$
$\text{coordinates of the ordered pairs in the relation contain no}$
$\text{repetitions. Scanning the first coordinates of the ordered pairs}$
$\text{of" \ \ R, \ "we see that none of them occur repeated. Each first}$
$\text{coordinate occurs only once !! Thus:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \setminus R \setminus \setminus \text{is a function.}$