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Describing Equivalence Classes

  1. Mar 6, 2010 #1
    Hey guys!
    So I am having trouble understanding equivalence classes. How are they determined??
    Anyways here is my problem!

    1. The problem statement, all variables and given/known data
    Let A and B be two sets, and f: A-->B a mapping. A relation on A is defined by: x~y iff f(x) = f(y)
    a) Show ~ is an equivalence relation
    b) Describe the equivalence classes when f is 1-1
    c) What can be said about f if ~ has only one equivalence class?

    2. Relevant equations

    3. The attempt at a solution

    a) I've already done this and understand it:
    reflexive: f(x)=f(x)
    symmetric: f(x) = f(y), f(y) = f(x)
    transitive f(x) = f(y) and f(y) = f(z) then f(x) = f(z)

    b) Okay here is where I am having trouble
    so if f is 1-1, it means f(x) = f(y) --> x = y
    Then would the equivalence class be something like all x that are in A which get mapped to f(x)?
    So [x] = { x ϵ A | f(x) = f(y) } = {x ϵ A | x = y } = {x}


    c) I don't know get the above question so I don't understand this one either...
  2. jcsd
  3. Mar 6, 2010 #2


    User Avatar
    Science Advisor

    Yes, for b, the equivalence classes are all "singleton sets"- each set contains only one member of A.

    For (c) it is exactly the opposite- all members of A are in the same equivalence class so for all x and y, f(x)= f(y)- f is a "constant function".
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