Is (0,0) A function or not?

1 Answer
Feb 25, 2018

"Yes, it is a function."

Explanation:

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

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

"Here are two different reasons" \ R \ "is a function."

"1) If" \ \ R \ \ "were a not a function, it would contain a first"
"coordinate, say" \ a, "that is paired with more than one second"
"coordinate, say" \ b_1 \ "and" \ b_2 \, \ "where" \ b_1 != b_2.

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

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

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

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

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