I am currently self-studying abstract algebra via the Abstract Algebra: Theory and Applications online textbook by Judson and Beezer, located here. My educational background is in electrical engineering (emphasis on computer engineering) and my professional background is software engineering. I'd like to eventually be able to study category theory and apply it to the software I write, but I quickly felt like I was in over my head when jumping straight to books on category theory. So, I backtracked to abstract algebra, which seems like a common jumping-off point for various more rigorous/specialized mathematical studies. My mathematical background through undergrad included vector calculus, linear algebra, and discrete mathematics courses.
I'm doing my best to fully absorb and "intuitionize" chapter one, "The Preliminaries," and have made my way to the Equivalence Relations and Partitions section, which defined equivalence relations as such:
An equivalence relation on a set $\mathit X$ is a relation $\mathit R \subset X \times X$ such that:
- $\mathit (x,x) \in R$ for all $\mathit x \in X$ (reflexive property)
- $\mathit (x,y) \in R$ implies $\mathit (y,x) \in R$ (symmetric property)
- $\mathit (x,y)$ and $\mathit (y,z) \in R$ imply $\mathit (x,z) \in R$ (transitive property)
The text provides a number of examples in which this definition can "generalize equality" over mathematical constructs I am familiar with. In particular, Example 1.22, which introduces a generalized notion of equality for two functions $\mathit f$ and $\mathit g$ that are differentiable in $\mathbb R$ by classifying them as equivalent if their derivatives are equivalent:
$$\mathit f \sim g$$
$$\text{if}$$
$$\mathit f\,'(x) = g\,'(x)$$
In this scenario, $\mathit f$ and $\mathit g$ might not be the same/equivalent function, but they share a commonality that meets the three conditions specified above. A physical example might be position functions of two cars with the same velocity but different starting positions. Their positions are not equivalent, but their velocities are, and there could be scenarios where we want to study a subset of cars that satisfy this equivalence relation.
However, that seemed like a more "complicated" example, involving derivatives and the real number line, and I wanted to know what equivalence relations would look like for "trivial" sets, like finite sets of integers.
So, I introduced a set $\mathit X$, defined as
$$\mathit X = \{1,2,3,4\}$$
with a cross-product $\mathit X \times X$, defined as
$$\mathit X \times X = \\ \{(1,1),(1,2),(1,3),(1,4)\\ (2,1),(2,2),(2,3),(2,4)\\ (3,1),(3,2),(3,3),(3,4)\\ (4,1),(4,2),(4,3),(4,4)\}$$
And then I started looking for relations $\mathit R$ that would constitute an equivalence relation. I found several, detailed below.
| Name | Equivalence Relation | Set Diagram | Cartesian Plot |
|---|---|---|---|
| $\mathit R_{0}$ | $\mathit R_{0} = \{(1,1),(2,2),(3,3),(4,4)\}$ |  |
 |
| $\mathit R_{1}$ | $\mathit R_{1} = \{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)\}$ |  |
 |
| $\mathit R_{2}$ | $\mathit R_{2} = \{(1,1),(2,2),(3,3),(4,4),(1,3),(3,1)\}$ |  |
 |
| $\mathit R_{3}$ | $\mathit R_{3} = \{(1,1),(2,2),(3,3),(4,4),(1,2),(2,3),(1,3),(2,1),(3,2),(3,1)\}$ |  |
 |
However, only one of these $\mathit R$ equivalence relations makes "intuitive sense" to me: $R_{0}$. Or rather, it's the only one that I know how to "state" in a notation that I am familiar with: functions (and yes, perhaps that is intentional, as $R_{0}$ is the only relation that also satisfies the conditions necessary for being a mapping/function). I can write the following:
$$\mathit x \sim y\,\text{if}\,x = y$$
$$\text{given that}$$
$$\mathit x \in X \,\text{and}\,y \in X$$
It's the "if x = y" bit that I'm hung up on. I'm reading the $\sim$ symbol as "equivalent," but that only makes intuitive sense to me if they're the same integer. Writing $1 \sim 1$ feels fine. But, it seems like there are scenarios where I can write $1 \sim 2$, which is messing with my brain.
For example, in $R_{1}$, it seems like I can say something like $1 \sim 2$, which certainly can't be expressed by the usual equals sign, $1 = 2$. Because one does not equal two! That being said, I can see some patterns in the set diagrams and Cartesian plots of these equivalence relations, and my gut tells me that that is where the notion of equivalence is hiding. But I don't know what those notions actually mean, beyond "this is a pattern, see?"
For the set diagrams, I feel like an expression like $\mathit a \sim b$ is saying something about traversability or routing. Like, $\mathit a \sim b$ implies that there is a path that traverses each $\mathit a$ and $\mathit b$ across both set diagrams (perhaps with the added notion that you never need to "backtrack" to hit all four spots, or that every "jump" is across sets rather than within them). There also seems to be a notion of symmetry, as every set diagram can be split down the middle and each side mirrors the other.
For the Cartesian plots, I feel like $\mathit a \sim b$ is saying something about the ways you can "pack" subsets into the full set. Or perhaps, it's saying something about "equivalent" (or maybe "similar") subsets. Here too there is a notion of symmetry about the $y=x$ axis.
But I don't know how to "express" any of this in a "mathematical" way. Am I just completely off in left field here? Is it the case that finite sets of integers are not at all "trivial" compared to differentiable functions and the real number line? I want to get some guidance before I go too deep on my own and end up with a bunch of faulty assumptions and intuitions about things.
It's best not to read "$\sim$" as "equivalent" because that doesn't mean anything; the meaningful concept is "equivalent for some purpose" as JonathanZ says in the comments. A nice and very useful example is an equivalence relation on the integers called $x \equiv y \bmod n$ (where $n$ is fixed), which means that $n$ divides $x - y$; this equivalence relation is the beginning of modular arithmetic. For example, if $x$ and $y$ are numbers of hours elapsed and you want to know what time it is, it's natural to take $n = 24$. Modular arithmetic is useful for many purposes, such as doing cryptography.
In full generality, you can show that placing an equivalence relation on a set is exactly the same thing as partitioning that set into disjoint subsets, known as the equivalence classes of the equivalence relation. So here are the equivalence relations you wrote down in terms of such partitions:
You can partition the elements of a set into equivalence classes however you want, but that doesn't necessarily mean the partition will be useful for anything. In mathematics we usually have some specific purpose in mind that we want to use some equivalence relation for.