Question about using the word unique

[Mr Davis 97]

I am trying to say that an element $a$ is paired with an element $b$ such that $b$ is paired with no other element.

I would like to write this more succinctly by just saying that $a$ is paired with a unique element $b$. However, it seems that this could also be interpreted as meaning that $a$ is paired with exactly one element $b$, while not necessarily implying that $b$ is not paired with any other element.

I need to get another opinion on what to do.

 

[mfb]

b is paired with a unique element a?

$\exists!$ x: x paired with b

 

[Mr Davis 97]

So does $a$ is paired with a unique $b$ mean that $a$ is associated with only one element, while $b$ is paired with a unique $a$ means that $a$ is paired with an element $b$ such that $b$ is paired with no other element?

 

[fresh_42]

Are $a$ and $b$ from different sets?
Can we distinguish $(a,b)$ and $(b,a)$?
Is $(a,b) \wedge (a,c)$ with $b \neq c$ possible?
Are all $(a,.)$ paired with some element?
Are all $(.,b)$ paired with some element?

I ask in order to find out, whether there can be established a function, or if it is just any relation.

 

[Mr Davis 97]

Are $a$ and $b$ from different sets?
Can we distinguish $(a,b)$ and $(b,a)$?
Is $(a,b) \wedge (a,c)$ with $b \neq c$ possible?
Are all $(a,.)$ paired with some element?
Are all $(.,b)$ paired with some element?

I ask in order to find out, whether there can be established a function, or if it is just any relation.

I guess you could say that it is a bijective function from a finite set to itself

 

[fresh_42]

I guess you could say that it is a bijective function from a finite set to itself

In this case you just gave yourself the answer. Why bothering any pairing if it is already 1:1? Just write $(a,f(a))$.

 

[mfb]

I guess you could say that it is a bijective function from a finite set to itself

A general one? This is usually called a permutation, and does not have to have clear pairs, because f(f(a)) does not have to be a.