Often, we want to define functions from to another set by giving their values on representatives. For example, we might wish to define a function from the set of families to the set of eye colors by defining .

However, this does not give a well-defined function, because we can choose different representatives and of the same family, and would give different values for the two representatives. For to be a function, it would need to be unambiguous.

Therefore, when defining a function in this way, it is important to check that if (in other words, if ) then .