I am talking about a relation that is what Wikipedia describes as [left-unique and right-unique](http://en.wikipedia.org/wiki/Relation_%28mathematics%29#Special_types_of_binary_relations). I never heard these terms before, but I have heard of the alternatives (injective and functional). The question is, *which terminology do you recommend*? Should I include short definitions? (The context is a text in the area of formal methods. I'm not sure if this helps.)


These are some trade-offs that I see:


* I think that *left-unique* and *right-unique* are not widely known, but I'm not sure at all.
* *functional* is overloaded
* *injective* sounds too fancy (subjective, of course)
* *left-unique* and *right-unique* are symmetric (good, of course)


**Edit:** It seems the question is unclear. Here are more details. I describe sets *X* and *Y* and then say:


1. now we must find an injective and functional relation between sets *X* and *Y* such that...
2. now we must find a left-unique and right unique relation between sets *X* and *Y*...


Which one do you recommend? What other information would you add? The relation does *not* have to be total. For example, various different ranges correspond to different 'feasible' relations. Technically I should not need to say that the relation does not have to be total, but will many people assume that it has to be total if I don't say it?



Do you need to say what left-unique and right-unique means?
Injective and functional are completely standard in this case. This is what you should use. The term "functional" is not overloaded, when you are using it to say that something is a function. Being functional means exactly that the relation is a function.


A relation that is injective and functional is precisely an injective function on its domain. It is a bijection of its domain with its range.


If you don't want to think of the relation as a function, then you can also describe it as a *one-to-one correspondence* of its domain with its range.


(And I don't think any of these terms I suggest would need to be defined, since their meaning is fairly universally known. This would definitely not be true of left-unique and right-unique.)


