easy homework: Equivalence classes, how do they look?

Question

Let's say that I got a

 set = { Arnold, Harrison }

and I want to display the equivalence class of [ Harrison ]

The actual condition for the relation doesn't matter in this case so let's just say that {Arnold} is the only relation to Harrison.

This would be displayed as:

[ Harrison ] = { Arnold }

However, if we have this set instead:

set = { Arnold, Harrison, {Arnold, Harrison}, { Arnold, Arnold, Harrison} }

And we say that this time, Harrison has a relation to Arnold but also {Arnold, Harrison} and even {Arnold, Arnold, Harrison}.

How is this expressed as an equivalence class?

[ Harrison ] = {Arnold, Harrison} 

or...

[ Harrison ] = { Arnold, {Arnold, Harrison}, {Arnold, Arnold, Harrison}  }

or... what?!

Answer 1

Your example (or "the homework text") doesn't seem too enlightening because all the elements in the set seem to be equivalent to each other.

Remember that the equivalence class of an element $s$ in a set $S$ with an equivalence relation $\sim$ is simply the subset of $S$ whose element are the elements of $S$ equivalent to $s$: $$ [s]=\{x\in S\,\mid\,x\sim s\}. $$ So, for instance, if $S=\{a,b,c,d,e\}$ and

• $a$ is equivalent only to $c$ and itself,
• $b$ is equivalent only to $e$ and itself,
• $d$ is equivalent only to itself,

then the equivalence classes are $[a]=[c]=\{a,c\}$, $[b]=[e]=\{b,e\}$ and $[d]=\{d\}$.

Furthermore, the quotient set is $S/\sim=\{[a],[b],[d]\}$.

Answer 2

Note that the equivalence class of Harrison must always contain Harrison. Otherwise, if I understand your relations correctly the second answer is closer to being correct.

[Harrison] = {Harrison, Arnold, {Arnold, Harrison}, {Arnold, Arnold, Harrison}}

Note that usually we do not repeat elements of a set, so that this can be more concisely written as

[Harrison] = {Harrison, Arnold, {Arnold, Harrison}}

Since {Arnold, Harrison} = {Arnold, Arnold, Harrison}