1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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".
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook