Partition of an equivalence relation

Question

I am having a hard time with the following problem:

In F(R), let f~g iff f(x)=g(x) for all x>c where c is some fixed real number.

I proved that it was a equivalence relation by the following:

1. f~f __ f(x)=f(x) so that is fine.

2. f~g implies g~f __ f(x)=g(x) and g(x)=f(x) f~g does imply g~f so that is fine.

3. f~g, g~h implies f~h __ f(x)=g(x) and g(x)=h(x) therefore you can substitute h(x) for g(x) thus f(x)=g(x).

The part that I am having trouble with is describing the partition associated with this equivalence relation. I know that the partition is equivalent to the equivalence class but I am unsure about how to find that since I'm dealing with functions. My assumption is that the partition is the collection of functions of f(x) where x>c.

Answer 1

I would say that for any function $f\in F(\mathbb R)$, the equivlence class containing $f$ (denoted $[f]$) is adequately described by $$[f]=\lbrace{g\in F(\mathbb R)}\mid g-f\equiv 0 \mbox{ on } (c,\infty)\}.$$ There are other equivalent ways to describe $[f]$; that's just one way.

Answer 2

If $c$ is a fixed constant, independent of $f$ and $g$, then the equivalence classes are essentially the functions with domain $(c,\infty)$. (That is, two functions are equivalent if and only if their restrictions to $(c,\infty)$ are identical.)

On the other hand, if $c=c(f,g)$, then you still have an equivalence relation, but the partition is different. In this case, two functions $f$ and $g$ are equivalent when they are "eventually equal": $[f]$ is the set of functions that are eventually (for large enough $x$) equal to $f$.

Answer 3

Your proofs are missing some detail:

• Reflexivity is obvious, as you noted.

• Symmetry is almost obvious: if $f\sim g$, then $f(x)=g(x)$ for all $x>c$; then $g(x)=f(x)$ for all $x>c$; therefore $g\sim f$.

• About transitivity, you miss a fact. Saying $f\sim g$ and $g\sim h$ means there exist $c$ and $d$ such that $f(x)=g(x)$ for all $x>c$ and $g(x)=h(x)$ for all $x>d$; however nothing allows you to assume $c=d$. But the argument can be easily fixed.

‘Describing’ the partition is a rather generic term.