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Just wondering if there exist any equivalence relations that don't actually involve some sort of equality between the pairs of elements in the equivalence relation set.

Equivalence relations involving some sort of equality are easy to construct. For example, the relation over the reals that relates all numbers that are some rational multiple of another number is an equivalence relation (reflexive by multiplication by 1, symmetric by reciprocal multiples, and transitive by multiples multiplied by multiples). Basically, if some fractional amount of some number is EQUAL to some other number, those two numbers can be regarded as equivalent in this perspective (interesting graph theory oriented observations here too).

Can we define some relation from and to some set of objects where two objects are "equivalent" with actually no characteristic in common? If there does not exist any such equivalence relation, then these relations were aptly named, unlike the "closed" and "open" properties of sets. And also, if such a relation doesn't exist, is there at least some equivalence relation that specifies equivalence between objects that are not intuitively equivalent?

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    $\begingroup$ Sure, all you need are the three properties (reflexive, symmetric, and transitive), the actual objects don't matter. You could define an equivalence relation on $\{x,4,\bullet,\square,\&,cat\}$. Now, whether this relation will be useful is another story. $\endgroup$ – Michael Burr May 22 '18 at 2:00
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In principle, an equivalence relation need only satisfy the axioms of an equivalence relation: reflexivity, symmetry, and transitivity. We can define relations on sets which satisfy these three axioms in any number of ways. For example, if I ask a room full of students to work in groups, I might assign the groups using some kind of random procedure. The groups themselves form equivalence classes under the relation "$A$ is related to $B$ if both students are in the same group." The only thing that two students in the same equivalence class have in common is that they are in the same equivalence class.

That being said, I think that you are going to find that most of the equivalence relations that turn up in Real Mathematics™ have an equality hidden somewhere. For example, a common topological construction of the Möbius band is as follows:

  • Start with a square.
  • Pick a distinguished side of the square.
  • Declare that a point on the distinguished side of the square is related to a point on the opposite side of the square if the segment joining the two points passes through the center of the square. This creates a set of classes each of which contains two points.
  • To ensure that the relation satisfies reflexivity, declare every singleton point to be related to itself.

Graphically, this can be represented via an image like the following: The Möbius band

Points on the red arrows are related to each other according to the orientation of the arrows. When we "quotient out" by this relation, the resulting space is the Möbius band. Note that I have never said that two points are related to each other if some quantities corresponding to those points are equal, hence I think that this is a useful and/or interesting example of the kind that you are looking for.

On the other hand, it might be easier to describe in coordinates using equality: Let $(x_1,y_1),(x_2,y_2) \in[0,1]^2$. We say that $(x_1,y_1)$ is related to $(x_2,y_2)$ if either

  • $(x_1, y_1) = (x_2,y_2)$,
  • $x_1 = 1-x_2$, $y_1 = 0$, and $y_2 = 1$, or
  • $x_1 = 1-x_2$, $y_1 = 1$, and $y_2 = 0$.

It is not too hard to check that this gives basically the same thing as the above, only in more notation and fewer words.

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Every equivalence relation may be defined by some kind of equality. To be more specific let $X$ be a set and $\mathcal{R}\subseteq X\times X$ be an equivalence relation in $X$. Given $x\in X$ let $[x]:=\{y\in X\,\mid\,(x,y)\in\mathcal{R}\}$ be the equivalence class of $x$ with respect to $\mathcal{R}$. Moreover let $X/\mathcal{R}:=\{[x]\,\mid\, x\in X\}$, the set of equivalence classes. Define $\pi\colon X\to X/\mathcal{R}$ by $\pi(x):=[x]$. Then $(x,y)\in\mathcal{R}$ iff $\pi(x)=\pi(y)$.

Moreover, given $X$ as above and given any nonempty set $Y$ and any function $p\colon X\to Y$ the set $\{(x,y)\in X\times X\,\mid\, p(x)=p(y)\}$ is an equivalence relation in $X$.

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An equivalence relation just has to satisfy reflexivity, symmetry, and transitivity.

Take the set $A=\{x,4,\bullet,\Box,cat\}$ in Michael Burr's comment. There are multiple equivalence relations definable on $A$, and the reason behind picking any one of them may be be arbitary or mathematical.

Here's something inbetween, taken from Efe Ok's Real Analysis with Economic Applications: Let $S$ be a nonempty set, and $R$ be a reflexive relation on $S$. The asymmetric part of $R$ is defined as the relation $P_R$ on $S$ as $xP_Ry$ iff $xRy$ but $y\not Rx$. Then the relation $I_R=R\setminus P_R$ on $S$ is called the symmetric part of $R$. Check that $I_R$ is reflexive and symmetric, and that if $R$ is transitive, then $I_R$ is transitive.
Now let $X$ be a nonempty set and $\succsim$ be a preference relation on $X$, defined as a preorder on $X$ (reflexive and transitive). The indifference relation $\sim$ is defined as the symmetric part of $\succsim$ and is an equivalence relation on $X$ by the above. So finally, we have an equivalence relation that is not completely arbitary (it's motivated by individual preferences) and may not be mathematical (after all, $X$ can be any set).

Ironically, for any set $X$, the diagonal relation $D_X=\{(x,x)\colon x\in X\}$ on $X$ is an equivalence relation so that $xD_X y$ iff $x$ is precisely $y$.

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