# What does unique mean?

1. Oct 24, 2007

### 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.

2. Oct 24, 2007

### 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.

3. Oct 25, 2007

### HallsofIvy

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

4. Oct 25, 2007

### waht

That makes sense, thanks.

5. Oct 25, 2007

### Owen Holden

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).

6. Oct 25, 2007

### HallsofIvy

Well, I'm glad we got that clarified!