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.
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: $f(0)=g(0) \ , \ g(0)=h(0) \Rightarrow f(0)=h(0)$.
What is $[\cos x]$ ? I'm not sure I understand the notation here...
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 $A/R$.
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.