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I Question about using the word unique

  1. Feb 25, 2017 #1
    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.
     
  2. jcsd
  3. Feb 25, 2017 #2

    mfb

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    2016 Award

    Staff: Mentor

    b is paired with a unique element a?

    ##\exists!## x: x paired with b
     
  4. Feb 25, 2017 #3
    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?
     
  5. Feb 25, 2017 #4

    fresh_42

    Staff: Mentor

    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.
     
  6. Feb 25, 2017 #5
    I guess you could say that it is a bijective function from a finite set to itself
     
  7. Feb 25, 2017 #6

    fresh_42

    Staff: Mentor

    In this case you just gave yourself the answer. Why bothering any pairing if it is already 1:1? Just write ##(a,f(a))##.
     
  8. Feb 25, 2017 #7

    mfb

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    2016 Award

    Staff: Mentor

    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.
     
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