I was working on #6. Once i find it is an equivalence relation how do i describe the partition?
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) ...
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)?