What is the difference between a well-defined relation and a function?

[HowToTrainYourDragon]

What exactly is the difference between saying a relation is well-defined vs. saying the relation is a function? Since a relation is well-defined iff each element of the domain is mapped to exactly one element in the codomain, aren't well-defined and function basically the same thing?

 

[jedishrfu]

Here's some discussion on it:

http://www.purplemath.com/modules/fcns.htm

 

[HallsofIvy]

What exactly is the difference between saying a relation is well-defined vs. saying the relation is a function? Since a relation is well-defined iff each element of the domain is mapped to exactly one element in the codomain, aren't well-defined and function basically the same thing?

No. That is not how "well defined" means. It means, rather, that you can get all the information required from the given definition. Yes, a "relation", as opposed to a "function" may have many "y" values corresponding to a given "x". But such a relation is "well defined" as long as it is possible to find all y corresponding to any given x.