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Can someone please explain in plain English what is meant by an equivalence class? I've been reading it over and over again and it just doesn't make any sense to me. I've tried looking up definitions online as well but they're too difficult to understand. If it's of any help, I know what an equivalence relation is and I know what congruence modulo are as well.

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here's an analogy that'll hopefully shed some light :

consider the equivalence relation 'lives in the same country' defined over the set of all the inhabitants of planet earth.

it is indeed an equivalence relation because :

  1. every person lives in the country that they live in
  2. if a person 'A' lives in the same country as a person 'B' then 'B' lives in the same country as 'A'
  3. if a person 'A' lives in the same country as a person 'B' and the person 'B' lives in the same country as a person 'C' then 'A' lives in the same country as 'C'

now think of the equivalence class of a person 'X' as their country

example : I'm Algerian my equivalence class with respect to that equivalence relation is simply Algeria

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When objects are "equivalent" it's a way of saying they are "the same". An equivalence class is a collections of objects that are all "equivalent" to each other, that is, we treat them all as being somehow the same object (using our equivalence relation to determine this).

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  • $\begingroup$ Thank you very much! I think it's when it was first introduced to me in congruence modulo that it became very confusing as there were multiple elements to the set. Idk, I'm just a slow person, but once I get it, I get it $\endgroup$ – user497020 Dec 28 '17 at 22:40
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    $\begingroup$ The above description may be used to argue that a subset of an equivalence class is also an equivalence class. I'd make it more precise to avoid this. $\endgroup$ – chi Dec 29 '17 at 13:01
  • $\begingroup$ @chi I was going for a "plain English" description, instead of getting technical. No one should use a description like this to argue anything. $\endgroup$ – Morgan Rodgers Dec 29 '17 at 15:16
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Given an equivalence relation $R$ on some set $S$, $R$ partitions $S$ in a very natural way. Each part of the partition is a subset of $S$, all elements of which are considered equivalent (e.i. related by $R$). Those are the equivalence classes.

For instance, let $S$ be some set of people, and let the relation be "has same colour shirt as". Then the equivalence classes will be "everyone with red shirt", "everyone with black shirt", "everyone with blue shirt" and so on.

You can even go the other way, first deciding what you want your equivalence classes should be, and then define the relation as "belongs to the same class as".

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As the name suggests, an equivalence class is a set of objects that are all in some sense 'equivalent' to each other.

For example we can take a bunch of objects and put all objects with the same color in an equivalence class. Or: having the same weight. Or same shape. Or ....

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An equivalence relation on a set $X$ is a partition of $X$, that is a way to divide $X$ into subsets which are non-empty and pairwise disjoint. Each of these subsets is an equivalence class.

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A classic equivalence relation and corresponding equivalence classes you've surely seen before is the set $S=\mathbb{Z}\times(\mathbb{Z}\backslash\{0\})$ with equivalence relation $(a,b)\sim(c,d)$ if and only if $ad=bc$, although you've most likely thought of $S$ as the rational numbers and $\sim$ as equality, i.e.:

$$\frac{a}{b}=\frac{c}{d}$$ if and only if $$ad=bc$$

An equivalence class is just a way of grouping objects that, though different, are the "same" in some way. $\frac{1}{3}$ and $\frac{3}{9}$ are, in fact, different under the above definition, but we treat them the same under this equivalence relation, as they belong to the same class.

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The analogy I always use is buckets.

The items in each bucket may be different, but up to some description are the same. Let's take people that live in certain states.

The bucket for Texas, for example, will include people from San Antonio, Austin, Houston, Bryan, McAllen... all over! But for the sake of classification, they're all Texans. They're all in the same equivalence class. The same can be said for a bucket (equivalence class) corresponding to Ohio: Canton, Dayton, Akron, Columbus... all people from Ohio!

Let's put it mathematically. The first example of equivalence classes people see is in modular arithmetic. Let's consider modulo 12 (the usual clock). Numbers with the same remainder upon division by 12 are placed in the same equivalence class - bucket - since they're essentially the same. In the class corresponding to 3, written $[3]_{12}$, includes $3, 15, 27,$ and so on. these numbers all have the same remainder when divided by 12.

In short - equivalence classes just classify objects that are, pretty much up to some descriptor, the same.

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Let $S$ be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$. For any two elements $a, b$ from $S$ let's say that $a\sim b$ if $a$ and $b$ give the same remainder when divided by $4$. It's not difficult to check that this relation is reflexive, symmetric, and transitive, so it is an equivalence relation. If $a\sim b$, we say $a$ and $b$ are equivalent.

Now let's split up $S$ into smaller subsets. In each subset, we will put all of the elements of $S$ which are equivalent to each other (under our definition of equivalent).

So the elements of $A=\{1, 5, 9 \}$ are equivalent to each other. Also the elements of $B=\{2, 6, 10\}$ are equivalent to each other, and the same can be said about $C=\{3, 7\}$ and $D=\{4, 8\}$. The sets $A, B, C, D$ are the equivalence classes induced on $S$ by the relation $\sim$. These sets have the proprety that they are all disjoint from each other, and if you unite all of them yo get back $S$. Therefore these sets are called a partition of $S$. Always, equivalence classes of any equivalence relation $\sim$ on a set $S$ partition $S$.

Now we could keep referring to these sets as $A, B, C, D$, which is convenient for this example (because these sets are easy to list explicitly). However, when the sets are not so small, we need a different method to name the equivalence classes. We might refer to $A$ as "the equivalence class which contains $5$", or, in short, "the equivalence class of $5$". Note that $A$ has $3$ possible names using this naming method: $A$ could be called the equivalence class of $1$, of $5$, or of $9$.

We name $B, C,$ and $D$ in a similar way.

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