I am currently reading about the subject given in the title of this thread. The definition they give for equivalence classes in my textbook is a rather ostentatious in its wording, so I just want to make certain that I am understanding it properly. They say to let R be an equivalence relation on a set A, meaning that this this particular relation is reflexive, symmetric, and transitive, right? Essentially the rest of it seems to say that you can partition off the elements that make the relation reflexive, thereby creating a subset of the relation R. Does that seem right?
I could really use some help, thank you!
Best Answer
Yes, any relation that satisfies these properties is by definition an equivalence relation. It is called an equivalence relation because it satisfies the fundamental properties of what it means for elements in a set to be "equal".
The important thing to understand is that it partitions up the set into disjoint (non-overlapping) subsets.
Let $X$ be a set of people standing in a crowded room, and define an equivalence relation $R$ on $X$ by saying that for any two people $x, y \in X$, $xRy$ if and only if $x$ and $y$ have first names beginning with the same letter. Then you divide up all the people in the room to non-overlapping subsets: $X_{a} \subset X$ people with first names beginning with $a$, $X_{b} \subset X$ beginning with $b$ , and so on.
Then you can write $X$ as the union of partitions defined by the relation $R$:
$$X = X_{a} \cup X_{b} \cup \dots \cup X_{y} \cup X_{z}.$$