Equivalence relation explanation I have started learning measure theory (Measure,Integral and Probability; Springer) and I am stuck/confused about page 4, where the book defines equivalence relation.
I am confused about what the $\sim$ symbol means. And especially in the case of reflexivity:
"[...] for any set E, an equivalence relation on E is a relation with the following properties:


*

*reflexive: for all $x \in E, x \sim x$ [...]"


Does $x \sim x$ mean that $(x,x) \in \mathbb{R}$ ? Why do we consider $2$-tuples if that is the case?
 A: Formally, a relation on $\mathbb R$ is just a subset of ${\mathbb R}^2.$ Some relations are functions (these are subsets of the plane such that no vertical line intersects the subset more than once), some relations are equivalence relations, some relations are partial orders, etc.
So technically speaking, to say that $\sim $ is an equivalence relation on $\mathbb R$ means that $\sim \; \subseteq {\mathbb R}^2$ and $\sim$ has some additional properties, such as the reflexive property, which means that $x \sim x$ for all real numbers $x$ (which technically means that $(x,x) \in \; \sim$ for all real numbers $x).$ The reflexive property holding for a relation means that the graph of $y=x$ is a subset of that relation. Thus, since any equivalence relation has the reflexive property, any equivalence relation in $\mathbb R$ will be a superset of the line defined by $y=x.$ Of course, not any superset of this line will be an equivalence relation, since there are two other conditions that also have to be satisfied (symmetry and transitivity).
Ordinary equality (i.e. $=)$ for real numbers is an example of an equivalence relation, and in this case the equivalence relation is EXACTLY the line defined by $y=x.$ That is, when viewing $=$ on the reals as a subset of the plane, that subset will be the line defined by $y=x.$ (Note that $=$ has the properties of symmetry and transitivity.) From what I've said, it follows that equality is the smallest equivalence relation.
Intuitively, equivalence relations are like "approximately equals" conditions. The larger an equivalence relation is, the further away from being equality it is, and thus the fuzzier the approximate equality is. The fuzziest of all the equivalence relations is all of ${\mathbb R}^2$ itself. Under this equivalence relation, every real number is equivalent to every other real number, which as you can imagine is not all that close to true equality!
