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The following definitions and results are from my textbook.

A partition $\mathcal{P}$ of a set $X$ is a collection of nonempty sets $X_1, X_2, \dots$ such that $X_1 \cap X_j = \emptyset$ for $i \neq j$ and $\bigcup_k X_k = X$. Let $\sim$ be an equivalence relation on a set $X$ and let $x \in X$. Then $[x] = \{y \in X: y \sim x\}$ is called the equivalence class of $x$. We will see that an equivalence relation gives rise to a partition via equivalence classes. Also, whenever a partition of a set exists, there is some natural underlying equivalence relation, as the following theorem demonstrates.

Theorem. Given an equivalence relation $\sim$ on a set $X$, the equivalence classes of $X$ form a partition of $X$. Conversely, if $\mathcal{P} = \{X_i\}$ is a partition of a set $X$, then there is an equivalence relation on $X$ with equivalence classes $X_i$.

Corollary. Two equivalence classes of an equivalence relation are either disjoint or equal.

This is great and all, but I do not really understand how to find the partitions in practice... can anyone help me? An intuitive explanation of why the partitions arise/the connection between partition, equivalence classes, and equivalence relations would also be very helpful. Thanks.

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    $\begingroup$ You might start by reading my answer and Tara B’s answer to this question; after that you might be able to make your question a little more precise/limited. $\endgroup$ – Brian M. Scott Sep 2 '15 at 23:52
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Equivalence relations (which are basically 'the same' as partitions by the Theorem) arise everywhere.

For a generic example, any function $f:A\to B$ determines an equivalence relation on $A$ by $a\sim a'$ iff $f(a)=f(a')$.

E.g. two persons are equivalent with respect to their height if they have the same height (here $f$ maps a person his/her height, say in integer millimeters). Then the partition (which consists of the equivalence classes) divides the people in as many groups as many possible heights are.

You can find infinitely many other examples.

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While Brian's link is as good of an answer as I would ever be able to give to the technical aspects, I would like to point out that equivalence relations and equivalence classes are one of the most common tools used in mathematics. We use them all the time to remove aspects of a construction we are not interested in, leaving only what we are after left behind.

A practical example from differential topology: Start with a square-shaped region of the plane, with vertices labeled A, B, C, D. This is an example of a "manifold with boundary", the boundary being the edges of the square. Our purpose to get rid of the boundary, leaving us with a compact (finite in size) manifold without boundary. We do this by building equivalence relations that preserve the structure we want (nearness) while getting rid of the boundary we don't.

I will produce 4 equivalence relations. Each gives us a different manifold. All four have this in common: every interior point of the square is equivalent only to itself. And every edge point is equivalent to exactly one other edge point, (except the vertices). How they differ is in how these points are stitched together. Denote the 4 sides by their endpoints: $AB, BC, CD, DA,$ or $BA, CB, DC, AD$. When I say that side $XY \sim WZ$, I mean that $X \sim W, Y \sim Z$ and any point on $XY$ a distance $t$ from $X$ will be equivalent to (and only to) the point on $WZ$ at a distance of $t$ from $W$.

  1. Relation (1) makes $AB \sim AD$ and $BC \sim DC$.
  2. Relation (2) makes $AB \sim DC$ and $AD \sim BC$.
  3. Relation (3) makes $AB \sim DC$ and $AD \sim CB$.
  4. Relation (4) makes $AB \sim CD$ and $AD \sim CB$.

For each relation, we form a new space out of the equivalence classes. The equivalence classes themselves are the points in our new manifold. Since in all relations, the interior points of the square are equivalent only to themselves, they each show up as a singleton set in our new space. Effectively, the interior points are unchanged. On the edges, the equivalent points are combined by the equivalence relation to form a single new point. These equivalence relations preserve the concept of "nearness" (which for simplicity's sake, I will leave as intuitive), so we can think of it as "bending the square over and gluing the edges together". Except that by using equivalence relations, we don't actually have to warp the square in space. We get what we need just from the logic of the relation.

Relation (1) gives us a surface that behaves exactly like a sphere. And for that reason, we call it the sphere.

Relation (2) gives us a surface that behaves exactly like a torus (surface of a donut shape).

Relation (3) gives us a surface called the Klein Bottle. It is unorientable (it has only one side, like a möbius strip), and does not exist in 3D Euclidean space, except as a self-intersecting surface.

Relation (4) gives us a surface called the Projective Plane. It is also unorientable and also does not exist in 3D Euclidean space, except by intersecting.

So by using these equivalence relations, I am able to construct these 4 surfaces in a form that allows me to see how they are related and how they differ. And to explore their properties by examining behavior on a square. Every subject in mathematics is replete with similar examples.

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The somewhat cynical answer is that we don't find them in practice. Defining equivalence relations is easy, so do that and use the theorem to say there must be partitions.

What does happen in practice is taking "representatives" from the partitions. We might have a partition $P$, and pick a member $x$ and call him the representative from that partition. Then we'd refer to $P$ as $[x]$ for simplicity. Often we care that some operation (maybe addition, maybe something scarier) acts nicely on $[x]$. So that if we have $x,y \in [x]$ and some function $f$, we really like it when $f(x)=f(y)$, then $f([x])=f([y])$. We never actually need to 'compute' or 'find' the partition.

To answer your last question we form partitions when we have 'too much stuff'. When forming the reals, we take all rational Cauchy sequences, which is way too much stuff. Then we quotient out 'similar sequences' (co-Cauchy) to get the reals.

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A partition is just a way to divide up all elements (of $X$) into "groups" (intuitively teams for something like a sport or game, formally it are sets), which are the equivalence classes. What determines if two elements $x$ and $y$ get into the same class you ask? They do when they are related to each other for the equivalence relation ($\sim$), i.e. when $x\sim y$.

First of all, one could write down the equivalence classes for an equivalence relation $\sim$ on $A$ (I use $A$ instead of $X$ from now on) in the following manner:

  1. Take an $a \in A$.
  2. Find all the elements $b\in A$ such that $a\sim b$.
  3. Then the set $X$ which contains $a$ and all such $b$'s is an equivalence relation. It is because every element in that set is in relation with every other element in that set, and none of the elements in the set are in relation with any element of $A$ outside of this set.
  4. Redefine $A$ as $A\setminus X$. If $A$ is now empty, you are done. Else, go back to step 1.

An example might clear up some things. Consider the set $A = \{1,2,3,4,5,6,7,8,9\}$. Then some equivalence relations you might think of are

  1. $a\sim_1 b$ when $a\equiv b \operatorname{mod} 2$, i.e. when $a$ and $b$ are both even, or when they are both odd. $\sim_1$ gives rise to the equivalence classes $O = \{1,3,5,7,9\}$ and $E = \{2,4,6,8\}$, or the odds and the evens respectively. We can see that $O$ and $E$ are a partition for $A$, because $A = O\cup E$ and $O\cap E = \emptyset$. Some visuals might help too. In the one I created, all elements of $A$ are present, and they are divided by bars to show the equivalence relations. $$\underbrace{13579}_\text{one equiv. class} \, \mid \, 2468$$ Note that this last bit is not common notation that I know of.
  2. $a\sim_2 b$ when $\lceil a/3\rceil = \lceil b/3 \rceil$. This gives rise to the equivalence classes $\{1,2,3\}$, $\{4,5,6\}$ and $\{7,8,9\}$. We can see that the union of the three sets is $A$, and the intersection of two of them is always empty; again we can conclude that the equivalence classes of $\sim_2$ are a partition of $A$. The visual I constructed before, here becomes: $$123\,\mid 456\,\mid 789$$

All of this only serves as a way to help grasp these concepts. I didn't talk about practical applications (e.g. projective geometry).

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