# Is (0,0) A function or not?

Feb 25, 2018

$\text{Yes, it is a function.}$

#### Explanation:

$\text{(I assume you mean the set of ordered pairs that contains only}$
$\text{the 1 pair given. If not -- please let me know !!!)}$

$\text{The question is:" \qquad \qquad "Is" \quad \ R \ = \ { (0, 0) \} \quad \ "a function "?}$

$\text{Here are two different reasons" \ R \ "is a function.}$

$\text{1) If" \ \ R \ \ "were a not a function, it would contain a first}$
$\text{coordinate, say" \ a, "that is paired with more than one second}$
$\text{coordinate, say" \ b_1 \ "and" \ b_2 \, \ "where} \setminus {b}_{1} \ne {b}_{2.}$

$\text{So then:" \qquad ( a ,b_1 ) \quad "and" \quad( a ,b_2 ) \quad \ "are two distinct points, and}$
$\text{they both belong to} \setminus R .$

 "But this is impossible --" \ R \ "contains only one point. "

$\text{So:" \qquad \qquad \qquad \qquad \ R \ = \ \ { (0, 0) \} \ \quad "must be a function.}$

$\text{2) The only first coordinate" \ R \ \ "has, is" \ \ 0, \ "and it occurs only}$
$\text{once. So there are no repetitions in the first coordinates.}$

$\text{Thus:" \qquad \qquad \qquad \qquad \qquad R \ = \ \ { (0, 0) \} \ quad "is a function.}$