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What does unique mean?

  1. Oct 24, 2007 #1
    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. jcsd
  3. Oct 24, 2007 #2
    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.
  4. Oct 25, 2007 #3


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    And, of course, "g= h" mean g(x)= h(x) for every x in the range of f.
  5. Oct 25, 2007 #4
    That makes sense, thanks.
  6. Oct 25, 2007 #5
    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).
  7. Oct 25, 2007 #6


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    Well, I'm glad we got that clarified!
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