What is the meaning of a unique inverse for a bijective function?

[waht]

What does "unique" mean?

I ran into a trivial exercise. If a function f is bijective, show that it has an inverse. That's easy. But then, the question goes: if f has an inverse, show that it is unique.

I'm not really sure what is meant by "unique." I would assume it is has to do with the function's one-to-one correspondence. That each element in the function is taken cared of (mapped) one at a time. Is this a good analogy? This is not homework by the way.

 

[d_leet]

It means there is only one inverse. In other words if a function, f, has inverses g and h, then g=h, and there is really only one inverse.

 

[HallsofIvy]

And, of course, "g= h" mean g(x)= h(x) for every x in the range of f.

 

[waht]

That makes sense, thanks.

 

[Owen Holden]

I ran into a trivial exercise. If a function f is bijective, show that it has an inverse. That's easy. But then, the question goes: if f has an inverse, show that it is unique.

I'm not really sure what is meant by "unique." I would assume it is has to do with the function's one-to-one correspondence. That each element in the function is taken cared of (mapped) one at a time. Is this a good analogy? This is not homework by the way.

x is unique means, there is one and only one thing that x is.

'The' in the particular, in the singular, is the meaning of 'unique'.

The definite article 'the' refers to that one and only x.

The x such that Fx, is that (unique) x which satisfies Fx.

That there is only one x which satisfies Fx is defined:
EyAx(x=y <-> Fx).

The unique x which is F has the property G, means, EyAx((x=y <-> Fx) & Gy).

 

[HallsofIvy]

Well, I'm glad we got that clarified!