We'll define on the set: $A=\Bbb R^{[0,1]}$ the relation $R$ by $fRg$ if $f(0)=g(0)$.
- Make sure it's an equivilence relation.
- What is $[\cos x]$ ?
- Describe all the equivalence classes and the quotient group $A/R$.
- Describe a subset of $A$, in it only one element exactly from each equivalence class (a system of representatives for the quotient group).
Well, first of all to make sure I understand the notation, $A=\Bbb R^{[0,1]}$ means the set of all the functions from the section $[0,1]$ to the reals right ? And the relation is when two functions are equal when $x=0$.
If it was from the reals to the section then some of these functions would be the trig functions, but the other way around I don't think it would be any elementary function.
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In order to prove this is an equivalence relation we need to check reflexivity, symmetry and transitivity, so for reflexivity I suppose it's enough to say that $f(0)=f(0)$, symmetry: $f(0)=g(0) \rightarrow g(0)=f(0)$ and transitivity: $f(0)=g(0) \ , \ g(0)=h(0) \Rightarrow f(0)=h(0)$.
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What is $[\cos x]$ ? I'm not sure I understand the notation here…
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I suppose the equivalence classes are functions that equal $i\in \Bbb R$ when $X=0$ so there are $i$ equivalence classes. I have no idea how to describe $A/R$.
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That's basically every function because there can always be $fRf$.
Please share your thoughts on what I did and how to solve this.
Thanks.
Best Answer
Your answer to part 1 is fine.
The notation means "functions from the interval $[0, 1]$ to the reals"; the interval $[0, 1]$ is the set of all real numbers between $0$ and $1$, inclusive.
For part 2, the notation $[\cos x]$ means "the equivalence class of the function $x \mapsto \cos x$. What functions are equivalent (under this relation) to cosine? Ones that have the same value at $x = 0$ as cosine, i.e., all functions $f$ with $f(0) = 1$.
You've observed that for each number $i \in \mathbb R$, there's an equivalence class (consisting of all functions that take the value $i$ at $x = 0$). So the set of equivalence classes is in 1-to-1 correspondence with the real numbers: the functions whose value at $0$ is $i$ correspond to the real number $i$.
For part 4, you need to find one element of each class. So a typical class is the one for $i = 3.5$. Can you think of a function whose value at $0$ is $3.5$? Sure, you can think of millions of them. But a particularly easy one is the constant function $f(x) = 3.5$. And indeed, the set of all constant functions has the property that each function is in a different equivalence class, and each equivalence class contains one of these functions.
You were 90% of the way there with your reasoning. Keep up the good work.