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Discrete math - equivalence relation

  1. Dec 24, 2008 #1
    Let A be a set. For every set B and total function f:A->B we define a relation R on A by R={(x,y) belonging to A*A:f(x)=f(y)}



    *belonging to - because i dont know how to make the symbole....


    Prove that f is one-to-one if and only if the equivalence classes of R are all singletones
     
  2. jcsd
  3. Dec 24, 2008 #2
    You can write TeX code by inserting [ tex] ... [\ tex] and putting the code in between (obviously without the spaces). The notation for element of is \in: [tex] x \in A [/tex].

    You can hit quote to see the code behind what I wrote.
     
  4. Dec 24, 2008 #3
    Have you made any attempt at this problem? It should follow easily from the definitions.
     
  5. Dec 25, 2008 #4

    MathematicalPhysicist

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    OK, here is one attempt.
    f is 1-1 iff for every x,y if f(x)=f(y) then x=y.

    The relation is defined by ~<=> x~y<=>f(x)=f(y), thus the equivalence class [x]={y|y~x}={y|f(x)=f(y)} if it's one to one it's evident that [x] is a singleton.
     
  6. Dec 25, 2008 #5

    MathematicalPhysicist

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    Gold Member

    Btw, it's not in the right place this thread of yours, it should be in the set theory section.
     
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