I don't think there is any special name for that.
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A function is a mapping from one set to another such that each element from the first set is mapped onto exactly one element from the second set.
Two. The set {x} has the subsets {} and {x}.
A set with only one element in it. The only proper subset of such a set is the null set.
If a function has an inverse then it is a bijection between two sets. Each element in the first set is mapped to one, and only one, element of the second set. Therefore, for each element in the second set there is one, and only one, element in the first set. The function and its inverse, both define the relationship between the same pairs of elements.
The only proper subset of a set comprising one element, is the null set.