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Currently I am trying to understand the equivalence class from the Herstein's book. And I am facing problem to understand the concept and also can't understand the theorem given in the page number 8. Book name is Topics in algebra, second edition.

Theorem 1.1.1. The distinct equivalence classes of an equivalence relation on $A$ provide us with a decomposition $o$f A as a union of mutually disjoint subsets. Conversely, given a decomposition of $A$ as a union of mutually disjoint, nonempty subsets, we can define an equivalence relation on $A$ for which these subsets are the distinct equivalence classes.

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    $\begingroup$ math.stackexchange.com/questions/517238/… $\endgroup$ – hHhh Jun 24 '15 at 17:07
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    $\begingroup$ Hernestein? I don't think I have that textbook handy. Maybe you should post a copy of the theorem you want explained. $\endgroup$ – Doug Chatham Jun 24 '15 at 17:07
  • $\begingroup$ As an example: take the integers $\mathbb{Z}$. Now, all even integers put into a class $[0]$ (which is formally just the set $\{...,-2,0,2,...\}$). All odd integers put in a class $[1]$. Now no integer is even and odd so these classes are disjoint. Furthermore all integers are either even or odd so every integer is in either $[0]$ or $[1]$ but not the other. The equivalence relation here is $a\sim b$ if $a,b$ have the same parity (evenness or oddness). $\endgroup$ – Eoin Jun 24 '15 at 17:12
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    $\begingroup$ Quoting the theorem might give you an opportunity to highlight exactly what you don't understand. For instance, you claim to have problems, but it is hard to guess exactly what you need. $\endgroup$ – user228113 Jun 24 '15 at 17:28
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    $\begingroup$ @Sid Is the theorem I have added to your post the theorem you are interested in? $\endgroup$ – Martin Sleziak Jun 24 '15 at 18:22
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A nice illustration of the concept is the following:

Think of $A$ as the collection of all people in the world. Think of an equivalence relation as "nationality", so two members of $A$ are equivalent in the sense that they have the same nationality (assuming for simplicity that everyone has unique nationality). This naturally decomposes $A$ into countries, i.e. collections of people grouped by nationality, and then the countries are precisely the equivalence classes. The theorem then reads:

The distinct countries provide us with a decomposition of all people of the world as a union of mutually disjoint groupings of people. Conversely, given a decomposition of the world as a union of mutually disjoint groupings of people, we can define an equivalence relation on the people of the world (= the nationality), for which the groupings by nationality are precisely the distinct countries.

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