I recently encountered what seems to me like an inconsistency in the usage of the term bijective. This is pretty basic stuff, but it somehow never occurred to me before. I'd like to make sure I'm using the term properly with respect to relations.
The definition (and, I assume, etymology) of bijective is usually given for functions as:
(1) injective and surjective.
But I've also heard the term used for relations. Thinking back, it would refer to a relation that is:
(2) injective and uniquely defined.
The point being that the bijective property should actually refer to the "one-to-one" nature of the relation or function in question.
(Functions get uniquely defined 'for free'. The extra ingredient for a bijective function is surjectivity, probably with the purpose that its inverse is then also a bijective function.)
But the two definitions are incompatible with the interpretation of functions as well-defined relations. Taking them both literally would result in contradiction: a non-surjective function would be bijective by (2), but not bijective by (1).
How is this usually dealt with? Is there a generally accepted alternative term for (2)?
Edit 1: As it turns out I am unable to find any reference to definition (2). I suppose my new question now becomes: Is there a nice term for uniquely defined, injective relations / injective partial functions?
Edit 2: Take a look at this Wikipedia section:
According to this, a relation can be bijective without being a function. It still doesn't correspond to (2) though. It keeps its etymological meaning: injective and surjective, i.e., the inverse of a function.
The nouns bijection, injection and surjection are reserved for functions, but the adjectives can apply to relations too.
(2) is called one-to-one there which, I suppose, is acceptable. :-)