1

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?!

Deragon
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2 Answers2

3

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 $S/\sim=\{[a],[b],[d]\}$.

William Phoenix
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Andrea Mori
  • 26,969
  • The text is made up but the structure is similar to the homework. If I were to modify your example to make it more like what I am confused about... Is if S also contained a subset {a, c}. If the condition for a relation to 'a' also would apply to that subset ( aR{a,c} ) ... Would [a] = {a, c, {a,c} in that case? – Deragon May 27 '13 at 11:06
  • If the set ${a,c}$ were itself an element of $S$, it'd be--ideally--an element in its own. So, other conditions on the equivalence aside, the fact that $a$ and $c$ are related says nothing about a relation between $a$ and ${a,c}$, or $c$ and ${a,c}$. For instance, the element $d$ in the set $S$ of my example could be actually ${a,c}$, and what follows would be exactly the same. – Andrea Mori May 27 '13 at 12:18
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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}

ronno
  • 11,390
  • What makes me confused is that I thought that you weren't allowed to have duplicates of same elements. {Harrison, Arnold, Arnold} for example would not be allowed right. But is {Harrison, Arnold, {Arnold, Harrison}} allowed then? I mean, both Arnold and Harrison are now mentioned twice in that equivalence class. – Deragon May 27 '13 at 10:50