# Examples of Equivalence Relations.

• Mar 21st 2012, 01:15 PM
Examples of Equivalence Relations.
http://i83.photobucket.com/albums/j306/xapharon/12B.jpg
I was working on #6. Once i find it is an equivalence relation how do i describe the partition?
• Mar 21st 2012, 01:36 PM
emakarov
Re: Examples of Equivalence Relations.
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I was working on #6.

What does $\displaystyle \mathcal{F}(\mathbb{R})$ denote?

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Once i find it is an equivalence relation how do i describe the partition?

By considering a lot of examples. Fix some f(x) and describe all functions equivalent to it, as well as functions not equivalent to it. What information needs to be provided to identify an equivalence class?
• Mar 21st 2012, 02:01 PM
Re: Examples of Equivalence Relations.
I think F(R) is set of all functions of real numbers. is it where f(x)=g(x) for some x in R?

To identify an equivalence class? well the equiv relation is f(0)=g(0), but that's all we know, so for every [f(0)]: f(0)=g(0) ...
• Mar 21st 2012, 02:09 PM
Plato
Re: Examples of Equivalence Relations.
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I think F(R) is set of all functions of real numbers. is it where f(x)=g(x) for some x in R?
To identify an equivalence class?

If $\displaystyle \mathcal{F}\left( \mathbb{R} \right)$ is the set of all real valued functions then every real number determines an equivalence class. WHY?
• Mar 21st 2012, 02:13 PM
Re: Examples of Equivalence Relations.
because every function f(x) is equal to another function g(x)? so f(x) = g(x) for every real number x?
• Mar 21st 2012, 02:50 PM
Plato
Re: Examples of Equivalence Relations.
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because every function f(x) is equal to another function g(x)? so f(x) = g(x) for every real number x?

Are you looking at #6?
As I read it $\displaystyle f\sim g$ if and only if $\displaystyle f(0)=g(0)$. Has nothing to do with any $\displaystyle x$.
If $\displaystyle c\in\mathbb{R}$ then define $\displaystyle [c]=\{f:f(0)=c\}$.
Can you show that the collection $\displaystyle \left\{ {[c]:c \in \mathbb{R}} \right\}$ partitions $\displaystyle \mathcal{F}(\mathbb{R})~.$
• Mar 21st 2012, 06:04 PM
Re: Examples of Equivalence Relations.
Yes, can you explain without using math talk. All i see are lines and squiggles.
• Mar 21st 2012, 06:09 PM
emakarov
Re: Examples of Equivalence Relations.
I would still consider examples.
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Originally Posted by emakarov
Fix some f(x) and describe all functions equivalent to it

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is it where f(x)=g(x) for some x in R?

As Plato said, no. Let, e.g., f(x) = x^2 + 1. How would you describe all functions equivalent to it? Non-equivalent to it?
• Mar 21st 2012, 06:15 PM
Re: Examples of Equivalence Relations.
Well there's the 'x' again. i thought plato said not to use an x. So now im getting even more confused

Anyway to describe the functions equivalent to f(x) is for some function g(x), f(x)=g(x) to be equivalent right?
and not equivalent f(x)≠g(x) right?

So what does this have to do with describing the partition?
• Mar 21st 2012, 06:21 PM
emakarov
Re: Examples of Equivalence Relations.
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Well there's the 'x' again. i thought plato said not to use an x.

Plato said that the definition of f ~ g in problem 6 does not use things like "for some x" or "for all x." In particular, f ~ g does not mean "f(x) = g(x) for every real number x," which is a quote from post #5.

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