I was working on #6. Once i find it is an equivalence relation how do i describe the partition?
What does $\displaystyle \mathcal{F}(\mathbb{R})$ denote?
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?
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) ...
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 $\displaystyle \mathcal{F}(\mathbb{R})~.$
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?
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.
This is not the definition of f ~ g from problem 6.
So, the question remains: describe all function g(x) such that g(x) ~ x^2 + 1.
how do we decide if f and g are equivalent? we compare f(0) and g(0). for example, if f(x) = x^2 + 1, then f(0) = 0^2 + 1 = 1, so if g ~ f, g(0) = 1. that's all we need to know, and in fact, it's all we CAN say about such functions g. so one way of characterizing [f] is saying "all g with the same y-intercept as f". how many different y-intercepts are there (that is, how many points do we have on the y-axis)?