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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...

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# Homework Help: Describing Equivalence Classes

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