How to find all the equivalence classes for a specific equivalence relation?

Question

What are the equivalence classes of the following equivalence relation $$S=\{(x,y) \in \mathbb{R} \times \mathbb{R} \mid x - y \in \mathbb{Q} \}$$?

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I know that an equivalence relation $R$ on a set $A$ induces a partition $P$ on that set. This partition $P$ contains all the equivalence classes of $A$ defined on the equivalence relation $R$.

An equivalence class is a set of that partition $P$, which contains sets that are pairwise disjoint (each element belongs exactly to just 1 set of the partition). All sets or equivalence classes of $P$ are non empty. The union of this sets of $P$ should be equal to $A$.

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That said, how would you go about searching the equivalence classes for the equivalence relation $S$ that I gave above as an example?

The notation for all equivalence classes should be something like this:

$$\mathbb{R}/S = \{ \text{equivalence classes} \}$$

How do I specify the equivalence classes, how can I find them?

I found also this notation:

$$[a]/S = \{ x \in R : aSx \}$$

which would mean the equivalence class where $[a]$ is: all $x$ in the same set of $a$ (in this case $\mathbb{R}$), such that $a$ is related to $x$ with the equivalence relation $S$.

Summarising, my problem consists of how to find and discover the equivalence classes for a equivalence relation $R$ defined on a $A$? What is the notation to represent them? How can I know it's correct?

Answer 1

You probably mean $R=\mathbb{R}$. The equivalence class that you define say that two elements $a,b\in \mathbb{R}$ are equivalent if and only if $a-b\in\mathbb{Q}$. Hence, the equivalence class of an element $a\in\mathbb{R}$ is equal to $\{a+x\mid x\in \mathbb{Q}\}$.

The set of equivalence classes is parametrised by $\mathbb{R}/\mathbb{Q}$, but I do not think that you can find an explicit family of representative elements.

Answer 2

Given an equivalence relation $S$ on a ground set $X$ the set of equivalence classes a priori is just an abstract set whose elements are subsets of $X$. This set of equivalence classes is usually denoted by $X/S$, or similar. When the equivalence of two elements $x$, $y\in X$ is written as $x\sim y$ (meaning the same thing as $(x,y)\in S$) then one would write $X/\!\sim$ for the set of equivalence classes. When this set occurs many times in the argument just introduce a new letter to denote it: "Let $\hat X:=X/\!\sim\>$".

How can the individual equivalence classes, being objects of a new type, be addressed? When there is just one equivalence relation at stake one writes $[a]$ for the equivalence class containing the element $a\in X$, being aware at all times that one and the same class is represented by many different elements of $X$; whence $[a]=[b]$ does not mean that $a=b$.

Sometimes each equivalence class contains a uniquely determined distinguished element. Then one can consider the set of these distinguished elements as a "model" of $X/\!\sim$. An example: Often ${\mathbb Q}$ is defined starting from "formal fractions" $a/b$ with $a\in{\mathbb Z}$, $\>b\in{\mathbb N}_{\geq1}$, for which a certain equivalence is defined. One then shows that each equivalence class contains a unique element with minimal $b\geq1$. This element is then the standard representant of the respective rational number.

In the example considered in your question it is impossible to find a distinguished representant for each class. It is therefore impossible to create an explicit "uncountable list" of these classes. If, however, you replace ${\mathbb R}/{\mathbb Q}$ by ${\mathbb R}/{\mathbb Z}$, i.e., consider two reals as equivalent when they differ by an integer, then the interval $[0,1[\ $ is a complete set of representants.